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Theorem ncfinprop 4474
 Description: Properties of finite cardinal number. Theorem X.1.23 of [Rosser] p. 527 (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
ncfinprop ((V Fin A V) → ( Ncfin A Nn A Ncfin A))

Proof of Theorem ncfinprop
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-ncfin 4442 . . . 4 Ncfin A = (℩x(x Nn A x))
2 vfinnc 4471 . . . . 5 ((A V V Fin ) → ∃!x Nn A x)
3 reiotacl 4364 . . . . 5 (∃!x Nn A x → (℩x(x Nn A x)) Nn )
42, 3syl 15 . . . 4 ((A V V Fin ) → (℩x(x Nn A x)) Nn )
51, 4syl5eqel 2437 . . 3 ((A V V Fin ) → Ncfin A Nn )
61eqcomi 2357 . . . 4 (℩x(x Nn A x)) = Ncfin A
7 eleq2 2414 . . . . . 6 (x = Ncfin A → (A xA Ncfin A))
87reiota2 4368 . . . . 5 (( Ncfin A Nn ∃!x Nn A x) → (A Ncfin A ↔ (℩x(x Nn A x)) = Ncfin A))
95, 2, 8syl2anc 642 . . . 4 ((A V V Fin ) → (A Ncfin A ↔ (℩x(x Nn A x)) = Ncfin A))
106, 9mpbiri 224 . . 3 ((A V V Fin ) → A Ncfin A)
115, 10jca 518 . 2 ((A V V Fin ) → ( Ncfin A Nn A Ncfin A))
1211ancoms 439 1 ((V Fin A V) → ( Ncfin A Nn A Ncfin A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!wreu 2616  Vcvv 2859  ℩cio 4337   Nn cnnc 4373   Fin cfin 4376   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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-ncfin 4442 This theorem is referenced by:  ncfindi  4475  ncfinsn  4476  ncfineleq  4477  ncfintfin  4495  vfinspnn  4541  1cvsfin  4542  vfintle  4546  vfin1cltv  4547  vfinncvntnn  4548  vfinspsslem1  4550  vfinncsp  4554  vinf  4555
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