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Theorem preqsn 4424
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 4223 . . 3 {𝐶} = {𝐶, 𝐶}
21eqeq2i 2663 . 2 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3 oridm 535 . . 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 4413 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)))
8 eqeq2 2662 . . . 4 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
98pm5.32ri 671 . . 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 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  {cpr 4212
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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-un 3612  df-sn 4211  df-pr 4213
This theorem is referenced by:  opeqsn  4996  propeqop  4999  propssopi  5000  relop  5305  hash2prde  13290  symg2bas  17864
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