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Theorem preqsn 4752
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (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 4749 . . 3 (𝐴 ∈ V → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
61, 5ax-mp 5 . 2 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶))
7 eqeq2 2810 . . 3 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
87pm5.32ri 579 . 2 ((𝐴 = 𝐵𝐵 = 𝐶) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
96, 8bitr4i 281 1 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  {csn 4525  {cpr 4527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528
This theorem is referenced by:  opeqsng  5358  propeqop  5362  propssopi  5363  relop  5685  hash2prde  13824  symg2bas  18513
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