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Theorem opeqpr 5511
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 2735 . 2 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ {𝐶, 𝐷} = ⟨𝐴, 𝐵⟩)
2 opeqpr.1 . . . 4 𝐴 ∈ V
3 opeqpr.2 . . . 4 𝐵 ∈ V
42, 3dfop 4877 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
54eqeq2i 2741 . 2 ({𝐶, 𝐷} = ⟨𝐴, 𝐵⟩ ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}})
6 opeqpr.3 . . 3 𝐶 ∈ V
7 opeqpr.4 . . 3 𝐷 ∈ V
8 snex 5437 . . 3 {𝐴} ∈ V
9 prex 5438 . . 3 {𝐴, 𝐵} ∈ V
106, 7, 8, 9preq12b 4856 . 2 ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
111, 5, 103bitri 296 1 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  Vcvv 3473  {csn 4632  {cpr 4634  cop 4638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639
This theorem is referenced by:  propeqop  5513  relop  5857
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