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Theorem preqsn 4366
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.)
Hypotheses
Ref Expression
preqsn.1 𝐴 ∈ V
preqsn.2 𝐵 ∈ V
preqsn.3 𝐶 ∈ V
Assertion
Ref Expression
preqsn ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Proof of Theorem preqsn
StepHypRef Expression
1 dfsn2 4166 . . 3 {𝐶} = {𝐶, 𝐶}
21eqeq2i 2633 . 2 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3 oridm 536 . . 3 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
4 preqsn.1 . . . 4 𝐴 ∈ V
5 preqsn.2 . . . 4 𝐵 ∈ V
6 preqsn.3 . . . 4 𝐶 ∈ V
74, 5, 6, 6preq12b 4355 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)))
8 eqeq2 2632 . . . 4 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
98pm5.32ri 669 . . 3 ((𝐴 = 𝐵𝐵 = 𝐶) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
103, 7, 93bitr4i 292 . 2 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
112, 10bitri 264 1 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  {csn 4153  {cpr 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-un 3564  df-sn 4154  df-pr 4156
This theorem is referenced by:  opeqsn  4932  propeqop  4935  propssopi  4936  relop  5237  hash2prde  13198  symg2bas  17750
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