Theorem eupth2lem2 28004

Description: Lemma for eupth2 28024. (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 2799	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → 𝐵 = 𝐵)
2	1	olcd 871	. . . . . 6 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))
3	2	biantrud 535	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 ≠ 𝐵 ↔ (𝐴 ≠ 𝐵 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵))))
4		eupth2lem2.1	. . . . . 6 ⊢ 𝐵 ∈ V
5		eupth2lem1 28003	. . . . . 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	syl6bbr 292	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 ≠ 𝐵 ↔ 𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
8		simpr 488	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → 𝐵 = 𝑈)
9	8	eleq1d 2874	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
10	7, 9	bitrd 282	. . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 ≠ 𝐵 ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
11	10	necon1bbid 3026	. 2 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝐴 = 𝐵))
12		simpl 486	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → 𝐵 ≠ 𝐶)
13		neeq1 3049	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐵 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶))
14	12, 13	syl5ibcom 248	. . . . . 6 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 → 𝐴 ≠ 𝐶))
15	14	pm4.71rd 566	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐴)))
16		eqcom 2805	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
17		ancom 464	. . . . 5 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐴))
18	15, 16, 17	3bitr4g 317	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶)))
19	12	neneqd 2992	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → ¬ 𝐵 = 𝐶)
20		biorf 934	. . . . . . 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 867	. . . . . 6 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐴) ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))
23	21, 22	syl6bb 290	. . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 = 𝐴 ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
24	23	anbi1d 632	. . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → ((𝐵 = 𝐴 ∧ 𝐴 ≠ 𝐶) ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶)))
25	18, 24	bitrd 282	. . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶)))
26		ancom 464	. . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) ↔ ((𝐵 = 𝐴 ∨ 𝐵 = 𝐶) ∧ 𝐴 ≠ 𝐶))
27	25, 26	syl6bbr 292	. 2 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐴 = 𝐵 ↔ (𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))))
28		eupth2lem1 28003	. . . 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 2874	. . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (𝐵 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
31	29, 30	bitr3id 288	. 2 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → ((𝐴 ≠ 𝐶 ∧ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
32	11, 27, 31	3bitrd 308	1 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243  ifcif 4425  {cpr 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528

This theorem is referenced by:  eupth2lem3lem4  28016