Theorem eupth2lem1 30876

Description: Lemma for eupth2 30897. (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 2853	. . 3 ⊢ (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
2	1	bibi1d 346	. 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 2853	. . 3 ⊢ ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
4	3	bibi1d 346	. 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 4286	. . . 4 ⊢ ¬ 𝑈 ∈ ∅
6	5	a1i 11	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7		simpl 488	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)) → 𝐴 ≠ 𝐵)
8	7	neneqd 2964	. . . 4 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9		simpr 490	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
10	8, 9	nsyl3 139	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ¬ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))
11	6, 10	2falsed 379	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
12		elprg 4607	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))
13		df-ne 2960	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
14		ibar 538	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ((𝑈 = 𝐴 ∨ 𝑈 = 𝐵) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
15	13, 14	sylbir 238	. . 3 ⊢ (¬ 𝐴 = 𝐵 → ((𝑈 = 𝐴 ∨ 𝑈 = 𝐵) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
16	12, 15	sylan9bb 519	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
17	2, 4, 11, 16	ifbothda 4524	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  ifcif 4485  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591

This theorem is used by:  eupth2lem2  30877  eupth2lem3lem6  30891