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Theorem ncfineleq 4477
 Description: Equality law for finite cardinality. Theorem X.1.24 of [Rosser] p. 527. (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
ncfineleq ((V Fin A V B W) → (A Ncfin BNcfin A = Ncfin B))

Proof of Theorem ncfineleq
StepHypRef Expression
1 simpl 443 . . . . 5 (((V Fin A V B W) A Ncfin B) → (V Fin A V B W))
2 ncfinprop 4474 . . . . . 6 ((V Fin A V) → ( Ncfin A Nn A Ncfin A))
323adant3 975 . . . . 5 ((V Fin A V B W) → ( Ncfin A Nn A Ncfin A))
4 simpl 443 . . . . 5 (( Ncfin A Nn A Ncfin A) → Ncfin A Nn )
51, 3, 43syl 18 . . . 4 (((V Fin A V B W) A Ncfin B) → Ncfin A Nn )
6 ncfinprop 4474 . . . . . . 7 ((V Fin B W) → ( Ncfin B Nn B Ncfin B))
763adant2 974 . . . . . 6 ((V Fin A V B W) → ( Ncfin B Nn B Ncfin B))
87simpld 445 . . . . 5 ((V Fin A V B W) → Ncfin B Nn )
98adantr 451 . . . 4 (((V Fin A V B W) A Ncfin B) → Ncfin B Nn )
103simprd 449 . . . . 5 ((V Fin A V B W) → A Ncfin A)
1110adantr 451 . . . 4 (((V Fin A V B W) A Ncfin B) → A Ncfin A)
12 simpr 447 . . . 4 (((V Fin A V B W) A Ncfin B) → A Ncfin B)
13 nnceleq 4430 . . . 4 ((( Ncfin A Nn Ncfin B Nn ) (A Ncfin A A Ncfin B)) → Ncfin A = Ncfin B)
145, 9, 11, 12, 13syl22anc 1183 . . 3 (((V Fin A V B W) A Ncfin B) → Ncfin A = Ncfin B)
1514ex 423 . 2 ((V Fin A V B W) → (A Ncfin BNcfin A = Ncfin B))
16 eleq2 2414 . . 3 ( Ncfin A = Ncfin B → (A Ncfin AA Ncfin B))
1710, 16syl5ibcom 211 . 2 ((V Fin A V B W) → ( Ncfin A = Ncfin BA Ncfin B))
1815, 17impbid 183 1 ((V Fin A V B W) → (A Ncfin BNcfin A = Ncfin B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2859   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: (None)
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