Theorem eupth2lem1 28561

Description: Lemma for eupth2 28582. (Contributed by Mario Carneiro, 8-Apr-2015.)

Assertion

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

Proof of Theorem eupth2lem1

Step	Hyp	Ref	Expression
1		eleq2 2828	. . 3 ⊢ (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
2	1	bibi1d 343	. 2 ⊢ (∅ = 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 2828	. . 3 ⊢ ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
4	3	bibi1d 343	. 2 ⊢ ({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 4269	. . . 4 ⊢ ¬ 𝑈 ∈ ∅
6	5	a1i 11	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7		simpl 482	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)) → 𝐴 ≠ 𝐵)
8	7	neneqd 2949	. . . 4 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9		simpr 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
10	8, 9	nsyl3 138	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ¬ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))
11	6, 10	2falsed 376	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
12		elprg 4587	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))
13		df-ne 2945	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
14		ibar 528	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ((𝑈 = 𝐴 ∨ 𝑈 = 𝐵) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
15	13, 14	sylbir 234	. . 3 ⊢ (¬ 𝐴 = 𝐵 → ((𝑈 = 𝐴 ∨ 𝑈 = 𝐵) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
16	12, 15	sylan9bb 509	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
17	2, 4, 11, 16	ifbothda 4502	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  ∅c0 4261  ifcif 4464  {cpr 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569

This theorem is referenced by:  eupth2lem2  28562  eupth2lem3lem6  28576