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Theorem preqsn 3537
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
preqsn.1 A V
preqsn.2 B V
preqsn.3 𝐶 V
Assertion
Ref Expression
preqsn ({A, B} = {𝐶} ↔ (A = B B = 𝐶))

Proof of Theorem preqsn
StepHypRef Expression
1 dfsn2 3381 . . 3 {𝐶} = {𝐶, 𝐶}
21eqeq2i 2047 . 2 ({A, B} = {𝐶} ↔ {A, B} = {𝐶, 𝐶})
3 preqsn.1 . . . 4 A V
4 preqsn.2 . . . 4 B V
5 preqsn.3 . . . 4 𝐶 V
63, 4, 5, 5preq12b 3532 . . 3 ({A, B} = {𝐶, 𝐶} ↔ ((A = 𝐶 B = 𝐶) (A = 𝐶 B = 𝐶)))
7 oridm 673 . . . 4 (((A = 𝐶 B = 𝐶) (A = 𝐶 B = 𝐶)) ↔ (A = 𝐶 B = 𝐶))
8 eqtr3 2056 . . . . . 6 ((A = 𝐶 B = 𝐶) → A = B)
9 simpr 103 . . . . . 6 ((A = 𝐶 B = 𝐶) → B = 𝐶)
108, 9jca 290 . . . . 5 ((A = 𝐶 B = 𝐶) → (A = B B = 𝐶))
11 eqtr 2054 . . . . . 6 ((A = B B = 𝐶) → A = 𝐶)
12 simpr 103 . . . . . 6 ((A = B B = 𝐶) → B = 𝐶)
1311, 12jca 290 . . . . 5 ((A = B B = 𝐶) → (A = 𝐶 B = 𝐶))
1410, 13impbii 117 . . . 4 ((A = 𝐶 B = 𝐶) ↔ (A = B B = 𝐶))
157, 14bitri 173 . . 3 (((A = 𝐶 B = 𝐶) (A = 𝐶 B = 𝐶)) ↔ (A = B B = 𝐶))
166, 15bitri 173 . 2 ({A, B} = {𝐶, 𝐶} ↔ (A = B B = 𝐶))
172, 16bitri 173 1 ({A, B} = {𝐶} ↔ (A = B B = 𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  opeqsn  3980  relop  4429
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