Theorem eupth2lem2 30191

Description: Lemma for eupth2 30211. (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 2732	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → 𝐵 = 𝐵)
2	1	olcd 874	. . . . . 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 30190	. . . . . 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 2816	. . . 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 2967	. 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 2990	. . . . . . 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 2738	. . . . 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 2933	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → ¬ 𝐵 = 𝐶)
20		biorf 936	. . . . . . 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 870	. . . . . 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 30190	. . . 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 2816	. . 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 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ∅c0 4278  ifcif 4470  {cpr 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3900  df-un 3902  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574

This theorem is referenced by:  eupth2lem3lem4  30203