MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opthpr Structured version   Visualization version   GIF version

Theorem opthpr 4774
Description: An unordered pair has the ordered pair property (compare opth 5359) under certain conditions. (Contributed by NM, 27-Mar-2007.)
Hypotheses
Ref Expression
preqr1.a 𝐴 ∈ V
preqr1.b 𝐵 ∈ V
preq12b.c 𝐶 ∈ V
preq12b.d 𝐷 ∈ V
Assertion
Ref Expression
opthpr (𝐴𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem opthpr
StepHypRef Expression
1 preqr1.a . . 3 𝐴 ∈ V
2 preqr1.b . . 3 𝐵 ∈ V
3 preq12b.c . . 3 𝐶 ∈ V
4 preq12b.d . . 3 𝐷 ∈ V
51, 2, 3, 4preq12b 4773 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
6 idd 24 . . . 4 (𝐴𝐷 → ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
7 df-ne 3014 . . . . . 6 (𝐴𝐷 ↔ ¬ 𝐴 = 𝐷)
8 pm2.21 123 . . . . . 6 𝐴 = 𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
97, 8sylbi 218 . . . . 5 (𝐴𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
109impd 411 . . . 4 (𝐴𝐷 → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
116, 10jaod 853 . . 3 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 = 𝐶𝐵 = 𝐷)))
12 orc 861 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
1311, 12impbid1 226 . 2 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
145, 13syl5bb 284 1 (𝐴𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  {cpr 4559
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
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  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-ne 3014  df-v 3494  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  brdom7disj  9941  brdom6disj  9942
  Copyright terms: Public domain W3C validator