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Theorem opeqpr 4175
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 2141 . 2 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ {𝐶, 𝐷} = ⟨𝐴, 𝐵⟩)
2 opeqpr.1 . . . 4 𝐴 ∈ V
3 opeqpr.2 . . . 4 𝐵 ∈ V
42, 3dfop 3704 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
54eqeq2i 2150 . 2 ({𝐶, 𝐷} = ⟨𝐴, 𝐵⟩ ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}})
6 opeqpr.3 . . 3 𝐶 ∈ V
7 opeqpr.4 . . 3 𝐷 ∈ V
82snex 4109 . . 3 {𝐴} ∈ V
9 prexg 4133 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
102, 3, 9mp2an 422 . . 3 {𝐴, 𝐵} ∈ V
116, 7, 8, 10preq12b 3697 . 2 ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
121, 5, 113bitri 205 1 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wo 697   = wceq 1331  wcel 1480  Vcvv 2686  {csn 3527  {cpr 3528  cop 3530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536
This theorem is referenced by:  relop  4689
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