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Theorem opeqpr 3964
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
opeqpr.1 A V
opeqpr.2 B V
opeqpr.3 𝐶 V
opeqpr.4 𝐷 V
Assertion
Ref Expression
opeqpr (⟨A, B⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {A} 𝐷 = {A, B}) (𝐶 = {A, B} 𝐷 = {A})))

Proof of Theorem opeqpr
StepHypRef Expression
1 eqcom 2024 . 2 (⟨A, B⟩ = {𝐶, 𝐷} ↔ {𝐶, 𝐷} = ⟨A, B⟩)
2 opeqpr.1 . . . 4 A V
3 opeqpr.2 . . . 4 B V
42, 3dfop 3522 . . 3 A, B⟩ = {{A}, {A, B}}
54eqeq2i 2032 . 2 ({𝐶, 𝐷} = ⟨A, B⟩ ↔ {𝐶, 𝐷} = {{A}, {A, B}})
6 opeqpr.3 . . 3 𝐶 V
7 opeqpr.4 . . 3 𝐷 V
8 snexgOLD 3909 . . . 4 (A V → {A} V)
92, 8ax-mp 7 . . 3 {A} V
10 prexgOLD 3920 . . . 4 ((A V B V) → {A, B} V)
112, 3, 10mp2an 404 . . 3 {A, B} V
126, 7, 9, 11preq12b 3515 . 2 ({𝐶, 𝐷} = {{A}, {A, B}} ↔ ((𝐶 = {A} 𝐷 = {A, B}) (𝐶 = {A, B} 𝐷 = {A})))
131, 5, 123bitri 195 1 (⟨A, B⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {A} 𝐷 = {A, B}) (𝐶 = {A, B} 𝐷 = {A})))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wo 616   = 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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