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Theorem prid1g 3465
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
Assertion
Ref Expression
prid1g (A 𝑉A {A, B})

Proof of Theorem prid1g
StepHypRef Expression
1 eqid 2037 . . 3 A = A
21orci 649 . 2 (A = A A = B)
3 elprg 3384 . 2 (A 𝑉 → (A {A, B} ↔ (A = A A = B)))
42, 3mpbiri 157 1 (A 𝑉A {A, B})
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242   wcel 1390  {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:  prid2g  3466  prid1  3467  preqr1g  3528  opth1  3964  en2lp  4232  acexmidlemcase  5450  m1expcl2  8911
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