Theorem eupth2lem2 30238

Description: Lemma for eupth2 30258. (Contributed by Mario Carneiro, 8-Apr-2015.)

Hypothesis

Ref	Expression
eupth2lem2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
eupth2lem2	⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Proof of Theorem eupth2lem2

Step	Hyp	Ref	Expression
1		eqidd 2738	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → 𝐵 = 𝐵)
2	1	olcd 875	. . . . . 6 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))
3	2	biantrud 531	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 ≠ 𝐵 ↔ (𝐴 ≠ 𝐵 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
4		eupth2lem2.1	. . . . . 6 ⊢ 𝐵 ∈ V
5		eupth2lem1 30237	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵)))
7	3, 6	bitr4di 289	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 ≠ 𝐵 ↔ 𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
8		simpr 484	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → 𝐵 = 𝑈)
9	8	eleq1d 2826	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
10	7, 9	bitrd 279	. . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 ≠ 𝐵 ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
11	10	necon1bbid 2980	. 2 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵))
12		simpl 482	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → 𝐵 ≠ 𝐶)
13		neeq1 3003	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐵 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶))
14	12, 13	syl5ibcom 245	. . . . . 6 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 → 𝐴 ≠ 𝐶))
15	14	pm4.71rd 562	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐴)))
16		eqcom 2744	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
17		ancom 460	. . . . 5 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐴))
18	15, 16, 17	3bitr4g 314	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶)))
19	12	neneqd 2945	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → ¬ 𝐵 = 𝐶)
20		biorf 937	. . . . . . 7 ⊢ (¬ 𝐵 = 𝐶 → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐴)))
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐴)))
22		orcom 871	. . . . . 6 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐴) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))
23	21, 22	bitrdi 287	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
24	23	anbi1d 631	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → ((𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶) ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶)))
25	18, 24	bitrd 279	. . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶)))
26		ancom 460	. . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶))
27	25, 26	bitr4di 289	. 2 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))))
28		eupth2lem1 30237	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ (𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))))
29	4, 28	ax-mp 5	. . 3 ⊢ (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ (𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
30	8	eleq1d 2826	. . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
31	29, 30	bitr3id 285	. 2 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → ((𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
32	11, 27, 31	3bitrd 305	1 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  ∅c0 4333  ifcif 4525  {cpr 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629

This theorem is referenced by:  eupth2lem3lem4  30250