Theorem preq12b 3532

Description: Equality relationship for 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
preq12b	⊢ ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))

Proof of Theorem preq12b

Step	Hyp	Ref	Expression
1		preq12b.1	. . . . . 6 ⊢ A ∈ V
2	1	prid1 3467	. . . . 5 ⊢ A ∈ {A, B}
3		eleq2 2098	. . . . 5 ⊢ ({A, B} = {𝐶, 𝐷} → (A ∈ {A, B} ↔ A ∈ {𝐶, 𝐷}))
4	2, 3	mpbii 136	. . . 4 ⊢ ({A, B} = {𝐶, 𝐷} → A ∈ {𝐶, 𝐷})
5	1	elpr 3385	. . . 4 ⊢ (A ∈ {𝐶, 𝐷} ↔ (A = 𝐶 ∨ A = 𝐷))
6	4, 5	sylib 127	. . 3 ⊢ ({A, B} = {𝐶, 𝐷} → (A = 𝐶 ∨ A = 𝐷))
7		preq1 3438	. . . . . . . 8 ⊢ (A = 𝐶 → {A, B} = {𝐶, B})
8	7	eqeq1d 2045	. . . . . . 7 ⊢ (A = 𝐶 → ({A, B} = {𝐶, 𝐷} ↔ {𝐶, B} = {𝐶, 𝐷}))
9		preq12b.2	. . . . . . . 8 ⊢ B ∈ V
10		preq12b.4	. . . . . . . 8 ⊢ 𝐷 ∈ V
11	9, 10	preqr2 3531	. . . . . . 7 ⊢ ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷)
12	8, 11	syl6bi 152	. . . . . 6 ⊢ (A = 𝐶 → ({A, B} = {𝐶, 𝐷} → B = 𝐷))
13	12	com12 27	. . . . 5 ⊢ ({A, B} = {𝐶, 𝐷} → (A = 𝐶 → B = 𝐷))
14	13	ancld 308	. . . 4 ⊢ ({A, B} = {𝐶, 𝐷} → (A = 𝐶 → (A = 𝐶 ∧ B = 𝐷)))
15		prcom 3437	. . . . . . 7 ⊢ {𝐶, 𝐷} = {𝐷, 𝐶}
16	15	eqeq2i 2047	. . . . . 6 ⊢ ({A, B} = {𝐶, 𝐷} ↔ {A, B} = {𝐷, 𝐶})
17		preq1 3438	. . . . . . . . 9 ⊢ (A = 𝐷 → {A, B} = {𝐷, B})
18	17	eqeq1d 2045	. . . . . . . 8 ⊢ (A = 𝐷 → ({A, B} = {𝐷, 𝐶} ↔ {𝐷, B} = {𝐷, 𝐶}))
19		preq12b.3	. . . . . . . . 9 ⊢ 𝐶 ∈ V
20	9, 19	preqr2 3531	. . . . . . . 8 ⊢ ({𝐷, B} = {𝐷, 𝐶} → B = 𝐶)
21	18, 20	syl6bi 152	. . . . . . 7 ⊢ (A = 𝐷 → ({A, B} = {𝐷, 𝐶} → B = 𝐶))
22	21	com12 27	. . . . . 6 ⊢ ({A, B} = {𝐷, 𝐶} → (A = 𝐷 → B = 𝐶))
23	16, 22	sylbi 114	. . . . 5 ⊢ ({A, B} = {𝐶, 𝐷} → (A = 𝐷 → B = 𝐶))
24	23	ancld 308	. . . 4 ⊢ ({A, B} = {𝐶, 𝐷} → (A = 𝐷 → (A = 𝐷 ∧ B = 𝐶)))
25	14, 24	orim12d 699	. . 3 ⊢ ({A, B} = {𝐶, 𝐷} → ((A = 𝐶 ∨ A = 𝐷) → ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶))))
26	6, 25	mpd 13	. 2 ⊢ ({A, B} = {𝐶, 𝐷} → ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))
27		preq12 3440	. . 3 ⊢ ((A = 𝐶 ∧ B = 𝐷) → {A, B} = {𝐶, 𝐷})
28		prcom 3437	. . . . 5 ⊢ {𝐷, B} = {B, 𝐷}
29	17, 28	syl6eq 2085	. . . 4 ⊢ (A = 𝐷 → {A, B} = {B, 𝐷})
30		preq1 3438	. . . 4 ⊢ (B = 𝐶 → {B, 𝐷} = {𝐶, 𝐷})
31	29, 30	sylan9eq 2089	. . 3 ⊢ ((A = 𝐷 ∧ B = 𝐶) → {A, B} = {𝐶, 𝐷})
32	27, 31	jaoi 635	. 2 ⊢ (((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)) → {A, B} = {𝐶, 𝐷})
33	26, 32	impbii 117	1 ⊢ ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {cpr 3368

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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by:  prel12  3533  opthpr  3534  preq12bg  3535  preqsn  3537  opeqpr  3981  preleq  4233