Theorem prel12 3542

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 3476	. . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		eleq2 2101	. . . 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 3477	. . . 4 ⊢ 𝐵 ∈ {𝐴, 𝐵}
7		eleq2 2101	. . . 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 3396	. . . 4 ⊢ (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
11		eqeq2 2049	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐷))
12	11	notbid 592	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐷))
13		orel2 645	. . . . . . . . . . 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 2049	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶))
19	18	notbid 592	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐶))
20		orel1 644	. . . . . . . . . . 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 700	. . . . . 6 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → ((𝐵 = 𝐷 ∨ 𝐵 = 𝐶) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
26	5	elpr 3396	. . . . . . 7 ⊢ (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
27		orcom 647	. . . . . . 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 3541	. . . . . 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 629   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {cpr 3376

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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by: (None)