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Theorem ncfineq 4473
 Description: Equality theorem for finite cardinality. (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
ncfineq (A = BNcfin A = Ncfin B)

Proof of Theorem ncfineq
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . . . 4 (A = B → (A xB x))
21anbi2d 684 . . 3 (A = B → ((x Nn A x) ↔ (x Nn B x)))
32iotabidv 4360 . 2 (A = B → (℩x(x Nn A x)) = (℩x(x Nn B x)))
4 df-ncfin 4442 . 2 Ncfin A = (℩x(x Nn A x))
5 df-ncfin 4442 . 2 Ncfin B = (℩x(x Nn B x))
63, 4, 53eqtr4g 2410 1 (A = BNcfin A = Ncfin B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ℩cio 4337   Nn cnnc 4373   Ncfin cncfin 4434 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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rex 2620  df-uni 3892  df-iota 4339  df-ncfin 4442 This theorem is referenced by:  tncveqnc1fin  4544  vfintle  4546  vfin1cltv  4547  vfinncsp  4554
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