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Theorem addcnul1 4452
 Description: Cardinal addition with the empty set. Theorem X.1.20, corollary 1 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.)
Assertion
Ref Expression
addcnul1 (A +c ) =

Proof of Theorem addcnul1
Dummy variables a b c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3564 . 2 ((A +c ) = a ¬ a (A +c ))
2 rex0 3563 . . . . 5 ¬ c ((bc) = a = (bc))
32a1i 10 . . . 4 (b A → ¬ c ((bc) = a = (bc)))
43nrex 2716 . . 3 ¬ b A c ((bc) = a = (bc))
5 eladdc 4398 . . 3 (a (A +c ) ↔ b A c ((bc) = a = (bc)))
64, 5mtbir 290 . 2 ¬ a (A +c )
71, 6mpgbir 1550 1 (A +c ) =
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   +c cplc 4375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-nul 3551  df-addc 4378 This theorem is referenced by:  addcnnul  4453  nulge  4456  tfinltfinlem1  4500  eventfin  4517  oddtfin  4518
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