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Metamath Proof Explorer

Theorem pexmidlem7N

Description: Lemma for pexmidN . Contradict pexmidlem6N . (Contributed by NM, 3-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses pexmidlem.l ⊢ ≤ = ( le ‘ 𝐾 )
pexmidlem.j ⊢ ∨ = ( join ‘ 𝐾 )
pexmidlem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
pexmidlem.p ⊢ + = ( +𝑃 ‘ 𝐾 )
pexmidlem.o ⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
pexmidlem.m ⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
Assertion pexmidlem7N ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝑀 ≠ 𝑋 )

Proof

Step Hyp Ref Expression
1 pexmidlem.l ⊢ ≤ = ( le ‘ 𝐾 )
2 pexmidlem.j ⊢ ∨ = ( join ‘ 𝐾 )
3 pexmidlem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 pexmidlem.p ⊢ + = ( +𝑃 ‘ 𝐾 )
5 pexmidlem.o ⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
6 pexmidlem.m ⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
7 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝐾 ∈ HL )
8 simpl3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝑝 ∈ 𝐴 )
9 8 snssd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → { 𝑝 } ⊆ 𝐴 )
10 simpl2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝑋 ⊆ 𝐴 )
11 3 4 sspadd2 ⊢ ( ( 𝐾 ∈ HL ∧ { 𝑝 } ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → { 𝑝 } ⊆ ( 𝑋 + { 𝑝 } ) )
12 7 9 10 11 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → { 𝑝 } ⊆ ( 𝑋 + { 𝑝 } ) )
13 vex ⊢ 𝑝 ∈ V
14 13 snss ⊢ ( 𝑝 ∈ ( 𝑋 + { 𝑝 } ) ↔ { 𝑝 } ⊆ ( 𝑋 + { 𝑝 } ) )
15 12 14 sylibr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝑝 ∈ ( 𝑋 + { 𝑝 } ) )
16 15 6 eleqtrrdi ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝑝 ∈ 𝑀 )
17 3 5 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )
18 7 10 17 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )
19 3 4 sspadd1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 ) → 𝑋 ⊆ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) )
20 7 10 18 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝑋 ⊆ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) )
21 simpr3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) )
22 20 21 ssneldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → ¬ 𝑝 ∈ 𝑋 )
23 nelne1 ⊢ ( ( 𝑝 ∈ 𝑀 ∧ ¬ 𝑝 ∈ 𝑋 ) → 𝑀 ≠ 𝑋 )
24 16 22 23 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → 𝑀 ≠ 𝑋 )