Theorem prel12 3516

Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Hypotheses

Ref	Expression
preq12b.1	⊢ A ∈ V
preq12b.2	⊢ B ∈ V
preq12b.3	⊢ 𝐶 ∈ V
preq12b.4	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
prel12	⊢ (¬ A = B → ({A, B} = {𝐶, 𝐷} ↔ (A ∈ {𝐶, 𝐷} ∧ B ∈ {𝐶, 𝐷})))

Proof of Theorem prel12

Step	Hyp	Ref	Expression
1		preq12b.1	. . . . 5 ⊢ A ∈ V
2	1	prid1 3450	. . . 4 ⊢ A ∈ {A, B}
3		eleq2 2083	. . . 4 ⊢ ({A, B} = {𝐶, 𝐷} → (A ∈ {A, B} ↔ A ∈ {𝐶, 𝐷}))
4	2, 3	mpbii 136	. . 3 ⊢ ({A, B} = {𝐶, 𝐷} → A ∈ {𝐶, 𝐷})
5		preq12b.2	. . . . 5 ⊢ B ∈ V
6	5	prid2 3451	. . . 4 ⊢ B ∈ {A, B}
7		eleq2 2083	. . . 4 ⊢ ({A, B} = {𝐶, 𝐷} → (B ∈ {A, B} ↔ B ∈ {𝐶, 𝐷}))
8	6, 7	mpbii 136	. . 3 ⊢ ({A, B} = {𝐶, 𝐷} → B ∈ {𝐶, 𝐷})
9	4, 8	jca 290	. 2 ⊢ ({A, B} = {𝐶, 𝐷} → (A ∈ {𝐶, 𝐷} ∧ B ∈ {𝐶, 𝐷}))
10	1	elpr 3368	. . . 4 ⊢ (A ∈ {𝐶, 𝐷} ↔ (A = 𝐶 ∨ A = 𝐷))
11		eqeq2 2031	. . . . . . . . . . . 12 ⊢ (B = 𝐷 → (A = B ↔ A = 𝐷))
12	11	notbid 579	. . . . . . . . . . 11 ⊢ (B = 𝐷 → (¬ A = B ↔ ¬ A = 𝐷))
13		orel2 632	. . . . . . . . . . 11 ⊢ (¬ A = 𝐷 → ((A = 𝐶 ∨ A = 𝐷) → A = 𝐶))
14	12, 13	syl6bi 152	. . . . . . . . . 10 ⊢ (B = 𝐷 → (¬ A = B → ((A = 𝐶 ∨ A = 𝐷) → A = 𝐶)))
15	14	com3l 75	. . . . . . . . 9 ⊢ (¬ A = B → ((A = 𝐶 ∨ A = 𝐷) → (B = 𝐷 → A = 𝐶)))
16	15	imp 115	. . . . . . . 8 ⊢ ((¬ A = B ∧ (A = 𝐶 ∨ A = 𝐷)) → (B = 𝐷 → A = 𝐶))
17	16	ancrd 309	. . . . . . 7 ⊢ ((¬ A = B ∧ (A = 𝐶 ∨ A = 𝐷)) → (B = 𝐷 → (A = 𝐶 ∧ B = 𝐷)))
18		eqeq2 2031	. . . . . . . . . . . 12 ⊢ (B = 𝐶 → (A = B ↔ A = 𝐶))
19	18	notbid 579	. . . . . . . . . . 11 ⊢ (B = 𝐶 → (¬ A = B ↔ ¬ A = 𝐶))
20		orel1 631	. . . . . . . . . . 11 ⊢ (¬ A = 𝐶 → ((A = 𝐶 ∨ A = 𝐷) → A = 𝐷))
21	19, 20	syl6bi 152	. . . . . . . . . 10 ⊢ (B = 𝐶 → (¬ A = B → ((A = 𝐶 ∨ A = 𝐷) → A = 𝐷)))
22	21	com3l 75	. . . . . . . . 9 ⊢ (¬ A = B → ((A = 𝐶 ∨ A = 𝐷) → (B = 𝐶 → A = 𝐷)))
23	22	imp 115	. . . . . . . 8 ⊢ ((¬ A = B ∧ (A = 𝐶 ∨ A = 𝐷)) → (B = 𝐶 → A = 𝐷))
24	23	ancrd 309	. . . . . . 7 ⊢ ((¬ A = B ∧ (A = 𝐶 ∨ A = 𝐷)) → (B = 𝐶 → (A = 𝐷 ∧ B = 𝐶)))
25	17, 24	orim12d 687	. . . . . 6 ⊢ ((¬ A = B ∧ (A = 𝐶 ∨ A = 𝐷)) → ((B = 𝐷 ∨ B = 𝐶) → ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶))))
26	5	elpr 3368	. . . . . . 7 ⊢ (B ∈ {𝐶, 𝐷} ↔ (B = 𝐶 ∨ B = 𝐷))
27		orcom 634	. . . . . . 7 ⊢ ((B = 𝐶 ∨ B = 𝐷) ↔ (B = 𝐷 ∨ B = 𝐶))
28	26, 27	bitri 173	. . . . . 6 ⊢ (B ∈ {𝐶, 𝐷} ↔ (B = 𝐷 ∨ B = 𝐶))
29		preq12b.3	. . . . . . 7 ⊢ 𝐶 ∈ V
30		preq12b.4	. . . . . . 7 ⊢ 𝐷 ∈ V
31	1, 5, 29, 30	preq12b 3515	. . . . . 6 ⊢ ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))
32	25, 28, 31	3imtr4g 194	. . . . 5 ⊢ ((¬ A = B ∧ (A = 𝐶 ∨ A = 𝐷)) → (B ∈ {𝐶, 𝐷} → {A, B} = {𝐶, 𝐷}))
33	32	ex 108	. . . 4 ⊢ (¬ A = B → ((A = 𝐶 ∨ A = 𝐷) → (B ∈ {𝐶, 𝐷} → {A, B} = {𝐶, 𝐷})))
34	10, 33	syl5bi 141	. . 3 ⊢ (¬ A = B → (A ∈ {𝐶, 𝐷} → (B ∈ {𝐶, 𝐷} → {A, B} = {𝐶, 𝐷})))
35	34	impd 242	. 2 ⊢ (¬ A = B → ((A ∈ {𝐶, 𝐷} ∧ B ∈ {𝐶, 𝐷}) → {A, B} = {𝐶, 𝐷}))
36	9, 35	impbid2 131	1 ⊢ (¬ A = B → ({A, B} = {𝐶, 𝐷} ↔ (A ∈ {𝐶, 𝐷} ∧ B ∈ {𝐶, 𝐷})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1374  Vcvv 2535  {cpr 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357

This theorem is referenced by: (None)