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Theorem opeqpr 5386
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
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 2825 . 2 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ {𝐶, 𝐷} = ⟨𝐴, 𝐵⟩)
2 opeqpr.1 . . . 4 𝐴 ∈ V
3 opeqpr.2 . . . 4 𝐵 ∈ V
42, 3dfop 4794 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
54eqeq2i 2831 . 2 ({𝐶, 𝐷} = ⟨𝐴, 𝐵⟩ ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}})
6 opeqpr.3 . . 3 𝐶 ∈ V
7 opeqpr.4 . . 3 𝐷 ∈ V
8 snex 5322 . . 3 {𝐴} ∈ V
9 prex 5323 . . 3 {𝐴, 𝐵} ∈ V
106, 7, 8, 9preq12b 4773 . 2 ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
111, 5, 103bitri 298 1 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  {cpr 4559  cop 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564
This theorem is referenced by:  propeqop  5388  relop  5714
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