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Metamath Proof Explorer

Theorem inf3lem4

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 for detailed description. (Contributed by NM, 29-Oct-1996)

Ref Expression
Hypotheses inf3lem.1 ⊢ 𝐺 = ( 𝑦 ∈ V ↦ { 𝑤 ∈ 𝑥 ∣ ( 𝑤 ∩ 𝑥 ) ⊆ 𝑦 } )
inf3lem.2 ⊢ 𝐹 = ( rec ( 𝐺 , ∅ ) ↾ ω )
inf3lem.3 ⊢ 𝐴 ∈ V
inf3lem.4 ⊢ 𝐵 ∈ V
Assertion inf3lem4 ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ⊊ ( 𝐹 ‘ suc 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 inf3lem.1 ⊢ 𝐺 = ( 𝑦 ∈ V ↦ { 𝑤 ∈ 𝑥 ∣ ( 𝑤 ∩ 𝑥 ) ⊆ 𝑦 } )
2 inf3lem.2 ⊢ 𝐹 = ( rec ( 𝐺 , ∅ ) ↾ ω )
3 inf3lem.3 ⊢ 𝐴 ∈ V
4 inf3lem.4 ⊢ 𝐵 ∈ V
5 1 2 3 4 inf3lem1 ⊢ ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) )
6 5 a1i ⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) ) )
7 1 2 3 4 inf3lem3 ⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ≠ ( 𝐹 ‘ suc 𝐴 ) ) )
8 6 7 jcad ⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) ∧ ( 𝐹 ‘ 𝐴 ) ≠ ( 𝐹 ‘ suc 𝐴 ) ) ) )
9 df-pss ⊢ ( ( 𝐹 ‘ 𝐴 ) ⊊ ( 𝐹 ‘ suc 𝐴 ) ↔ ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) ∧ ( 𝐹 ‘ 𝐴 ) ≠ ( 𝐹 ‘ suc 𝐴 ) ) )
10 8 9 imbitrrdi ⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ⊊ ( 𝐹 ‘ suc 𝐴 ) ) )