Theorem eupath2lem1 26504

Description: Lemma for eupath2 26507. (Contributed by Mario Carneiro, 8-Apr-2015.)

Assertion

Ref	Expression
eupath2lem1	⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))

Proof of Theorem eupath2lem1

Step	Hyp	Ref	Expression
1		eleq2 2677	. . 3 ⊢ (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
2	1	bibi1d 332	. 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 2677	. . 3 ⊢ ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
4	3	bibi1d 332	. 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 3878	. . . 4 ⊢ ¬ 𝑈 ∈ ∅
6	5	a1i 11	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7		simpl 472	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)) → 𝐴 ≠ 𝐵)
8	7	neneqd 2787	. . . 4 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9		simpr 476	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
10	8, 9	nsyl3 132	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → ¬ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))
11	6, 10	2falsed 365	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
12		elprg 4144	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))
13		df-ne 2782	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
14		ibar 524	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ((𝑈 = 𝐴 ∨ 𝑈 = 𝐵) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
15	13, 14	sylbir 224	. . 3 ⊢ (¬ 𝐴 = 𝐵 → ((𝑈 = 𝐴 ∨ 𝑈 = 𝐵) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
16	12, 15	sylan9bb 732	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))
17	2, 4, 11, 16	ifbothda 4073	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ifcif 4036  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128

This theorem is referenced by:  eupath2lem2  26505  eupath2lem3  26506  eupth2lem2  41387  eupth2lem3lem6  41401