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Theorem opeqsn 3963
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
StepHypRef Expression
1 opeqsn.1 . . . 4 A V
2 opeqsn.2 . . . 4 B V
31, 2dfop 3522 . . 3 A, B⟩ = {{A}, {A, B}}
43eqeq1i 2029 . 2 (⟨A, B⟩ = {𝐶} ↔ {{A}, {A, B}} = {𝐶})
5 snexgOLD 3909 . . . 4 (A V → {A} V)
61, 5ax-mp 7 . . 3 {A} V
7 prexgOLD 3920 . . . 4 ((A V B V) → {A, B} V)
81, 2, 7mp2an 404 . . 3 {A, B} V
9 opeqsn.3 . . 3 𝐶 V
106, 8, 9preqsn 3520 . 2 ({{A}, {A, B}} = {𝐶} ↔ ({A} = {A, B} {A, B} = 𝐶))
11 eqcom 2024 . . . . 5 ({A} = {A, B} ↔ {A, B} = {A})
121, 2, 1preqsn 3520 . . . . 5 ({A, B} = {A} ↔ (A = B B = A))
13 eqcom 2024 . . . . . . 7 (B = AA = B)
1413anbi2i 433 . . . . . 6 ((A = B B = A) ↔ (A = B A = B))
15 anidm 376 . . . . . 6 ((A = B A = B) ↔ A = B)
1614, 15bitri 173 . . . . 5 ((A = B B = A) ↔ A = B)
1711, 12, 163bitri 195 . . . 4 ({A} = {A, B} ↔ A = B)
1817anbi1i 434 . . 3 (({A} = {A, B} {A, B} = 𝐶) ↔ (A = B {A, B} = 𝐶))
19 dfsn2 3364 . . . . . . 7 {A} = {A, A}
20 preq2 3422 . . . . . . 7 (A = B → {A, A} = {A, B})
2119, 20syl5req 2067 . . . . . 6 (A = B → {A, B} = {A})
2221eqeq1d 2030 . . . . 5 (A = B → ({A, B} = 𝐶 ↔ {A} = 𝐶))
23 eqcom 2024 . . . . 5 ({A} = 𝐶𝐶 = {A})
2422, 23syl6bb 185 . . . 4 (A = B → ({A, B} = 𝐶𝐶 = {A}))
2524pm5.32i 430 . . 3 ((A = B {A, B} = 𝐶) ↔ (A = B 𝐶 = {A}))
2618, 25bitri 173 . 2 (({A} = {A, B} {A, B} = 𝐶) ↔ (A = B 𝐶 = {A}))
274, 10, 263bitri 195 1 (⟨A, B⟩ = {𝐶} ↔ (A = B 𝐶 = {A}))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  {csn 3350  {cpr 3351  cop 3353
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 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-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  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-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359
This theorem is referenced by:  relop  4413
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