Theorem opeqsn 3980

Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Hypotheses

Ref	Expression
opeqsn.1	⊢ A ∈ V
opeqsn.2	⊢ B ∈ V
opeqsn.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
opeqsn	⊢ (⟨A, B⟩ = {𝐶} ↔ (A = B ∧ 𝐶 = {A}))

Proof of Theorem opeqsn

Step	Hyp	Ref	Expression
1		opeqsn.1	. . . 4 ⊢ A ∈ V
2		opeqsn.2	. . . 4 ⊢ B ∈ V
3	1, 2	dfop 3539	. . 3 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
4	3	eqeq1i 2044	. 2 ⊢ (⟨A, B⟩ = {𝐶} ↔ {{A}, {A, B}} = {𝐶})
5		snexgOLD 3926	. . . 4 ⊢ (A ∈ V → {A} ∈ V)
6	1, 5	ax-mp 7	. . 3 ⊢ {A} ∈ V
7		prexgOLD 3937	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
8	1, 2, 7	mp2an 402	. . 3 ⊢ {A, B} ∈ V
9		opeqsn.3	. . 3 ⊢ 𝐶 ∈ V
10	6, 8, 9	preqsn 3537	. 2 ⊢ ({{A}, {A, B}} = {𝐶} ↔ ({A} = {A, B} ∧ {A, B} = 𝐶))
11		eqcom 2039	. . . . 5 ⊢ ({A} = {A, B} ↔ {A, B} = {A})
12	1, 2, 1	preqsn 3537	. . . . 5 ⊢ ({A, B} = {A} ↔ (A = B ∧ B = A))
13		eqcom 2039	. . . . . . 7 ⊢ (B = A ↔ A = B)
14	13	anbi2i 430	. . . . . 6 ⊢ ((A = B ∧ B = A) ↔ (A = B ∧ A = B))
15		anidm 376	. . . . . 6 ⊢ ((A = B ∧ A = B) ↔ A = B)
16	14, 15	bitri 173	. . . . 5 ⊢ ((A = B ∧ B = A) ↔ A = B)
17	11, 12, 16	3bitri 195	. . . 4 ⊢ ({A} = {A, B} ↔ A = B)
18	17	anbi1i 431	. . 3 ⊢ (({A} = {A, B} ∧ {A, B} = 𝐶) ↔ (A = B ∧ {A, B} = 𝐶))
19		dfsn2 3381	. . . . . . 7 ⊢ {A} = {A, A}
20		preq2 3439	. . . . . . 7 ⊢ (A = B → {A, A} = {A, B})
21	19, 20	syl5req 2082	. . . . . 6 ⊢ (A = B → {A, B} = {A})
22	21	eqeq1d 2045	. . . . 5 ⊢ (A = B → ({A, B} = 𝐶 ↔ {A} = 𝐶))
23		eqcom 2039	. . . . 5 ⊢ ({A} = 𝐶 ↔ 𝐶 = {A})
24	22, 23	syl6bb 185	. . . 4 ⊢ (A = B → ({A, B} = 𝐶 ↔ 𝐶 = {A}))
25	24	pm5.32i 427	. . 3 ⊢ ((A = B ∧ {A, B} = 𝐶) ↔ (A = B ∧ 𝐶 = {A}))
26	18, 25	bitri 173	. 2 ⊢ (({A} = {A, B} ∧ {A, B} = 𝐶) ↔ (A = B ∧ 𝐶 = {A}))
27	4, 10, 26	3bitri 195	1 ⊢ (⟨A, B⟩ = {𝐶} ↔ (A = B ∧ 𝐶 = {A}))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368  ⟨cop 3370

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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  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-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376

This theorem is referenced by:  relop  4429