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Theorem opeqpr 4997
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 2658 . 2 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ {𝐶, 𝐷} = ⟨𝐴, 𝐵⟩)
2 opeqpr.1 . . . 4 𝐴 ∈ V
3 opeqpr.2 . . . 4 𝐵 ∈ V
42, 3dfop 4432 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
54eqeq2i 2663 . 2 ({𝐶, 𝐷} = ⟨𝐴, 𝐵⟩ ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}})
6 opeqpr.3 . . 3 𝐶 ∈ V
7 opeqpr.4 . . 3 𝐷 ∈ V
8 snex 4938 . . 3 {𝐴} ∈ V
9 prex 4939 . . 3 {𝐴, 𝐵} ∈ V
106, 7, 8, 9preq12b 4413 . 2 ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
111, 5, 103bitri 286 1 (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  {cpr 4212  cop 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217
This theorem is referenced by:  propeqop  4999  relop  5305
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