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

Theorem preqsn 4613
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof revised by AV, 12-Jun-2022.)
Hypotheses
Ref Expression
preqsn.1 𝐴 ∈ V
preqsn.2 𝐵 ∈ V
Assertion
Ref Expression
preqsn ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Proof of Theorem preqsn
StepHypRef Expression
1 preqsn.1 . . 3 𝐴 ∈ V
2 id 22 . . . 4 (𝐴 ∈ V → 𝐴 ∈ V)
3 preqsn.2 . . . . 5 𝐵 ∈ V
43a1i 11 . . . 4 (𝐴 ∈ V → 𝐵 ∈ V)
52, 4preqsnd 4609 . . 3 (𝐴 ∈ V → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
61, 5ax-mp 5 . 2 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶))
7 eqeq2 2836 . . 3 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
87pm5.32ri 571 . 2 ((𝐴 = 𝐵𝐵 = 𝐶) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
96, 8bitr4i 270 1 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1656  wcel 2164  Vcvv 3414  {csn 4399  {cpr 4401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-un 3803  df-nul 4147  df-sn 4400  df-pr 4402
This theorem is referenced by:  opeqsng  5189  opeqsnOLD  5191  propeqop  5195  propssopi  5196  relop  5509  hash2prde  13548  symg2bas  18175
  Copyright terms: Public domain W3C validator