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Metamath Proof Explorer

Theorem eupth2lem1

Description: Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015)

Ref Expression
Assertion eupth2lem1 ( 𝑈 ∈ 𝑉 → ( 𝑈 ∈ if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 eleq2 ⊢ ( ∅ = if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) → ( 𝑈 ∈ ∅ ↔ 𝑈 ∈ if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) ) )
2 1 bibi1d ⊢ ( ∅ = if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) → ( ( 𝑈 ∈ ∅ ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) ↔ ( 𝑈 ∈ if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) ) )
3 eleq2 ⊢ ( { 𝐴 , 𝐵 } = if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) → ( 𝑈 ∈ { 𝐴 , 𝐵 } ↔ 𝑈 ∈ if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) ) )
4 3 bibi1d ⊢ ( { 𝐴 , 𝐵 } = if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) → ( ( 𝑈 ∈ { 𝐴 , 𝐵 } ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) ↔ ( 𝑈 ∈ if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) ) )
5 noel ⊢ ¬ 𝑈 ∈ ∅
6 5 a1i ⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → ¬ 𝑈 ∈ ∅ )
7 simpl ⊢ ( ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) → 𝐴 ≠ 𝐵 )
8 7 neneqd ⊢ ( ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) → ¬ 𝐴 = 𝐵 )
9 simpr ⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 )
10 8 9 nsyl3 ⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → ¬ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) )
11 6 10 2falsed ⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → ( 𝑈 ∈ ∅ ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) )
12 elprg ⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∈ { 𝐴 , 𝐵 } ↔ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) )
13 df-ne ⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
14 ibar ⊢ ( 𝐴 ≠ 𝐵 → ( ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) )
15 13 14 sylbir ⊢ ( ¬ 𝐴 = 𝐵 → ( ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) )
16 12 15 sylan9bb ⊢ ( ( 𝑈 ∈ 𝑉 ∧ ¬ 𝐴 = 𝐵 ) → ( 𝑈 ∈ { 𝐴 , 𝐵 } ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) )
17 2 4 11 16 ifbothda ⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∈ if ( 𝐴 = 𝐵 , ∅ , { 𝐴 , 𝐵 } ) ↔ ( 𝐴 ≠ 𝐵 ∧ ( 𝑈 = 𝐴 ∨ 𝑈 = 𝐵 ) ) ) )