Theorem eupth2lem1 30247

Description: Lemma for eupth2 30268. (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 4344	. . . 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 2943	. . . 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 4653	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))
13		df-ne 2939	. . . 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 235	. . 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 4569	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∅c0 4339  ifcif 4531  {cpr 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634

This theorem is referenced by:  eupth2lem2  30248  eupth2lem3lem6  30262