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 e. Fin /\ A e. V /\ B e. W) -> (A e. Ncfin B <-> Ncfin A = Ncfin B))

Proof of Theorem ncfineleq

Step	Hyp	Ref	Expression
1		simpl 443	. . . . 5 |- (((_V e. Fin /\ A e. V /\ B e. W) /\ A e. Ncfin B) -> (_V e. Fin /\ A e. V /\ B e. W))
2		ncfinprop 4474	. . . . . 6 |- ((_V e. Fin /\ A e. V) -> ( Ncfin A e. Nn /\ A e. Ncfin A))
3	2	3adant3 975	. . . . 5 |- ((_V e. Fin /\ A e. V /\ B e. W) -> ( Ncfin A e. Nn /\ A e. Ncfin A))
4		simpl 443	. . . . 5 |- (( Ncfin A e. Nn /\ A e. Ncfin A) -> Ncfin A e. Nn )
5	1, 3, 4	3syl 18	. . . 4 |- (((_V e. Fin /\ A e. V /\ B e. W) /\ A e. Ncfin B) -> Ncfin A e. Nn )
6		ncfinprop 4474	. . . . . . 7 |- ((_V e. Fin /\ B e. W) -> ( Ncfin B e. Nn /\ B e. Ncfin B))
7	6	3adant2 974	. . . . . 6 |- ((_V e. Fin /\ A e. V /\ B e. W) -> ( Ncfin B e. Nn /\ B e. Ncfin B))
8	7	simpld 445	. . . . 5 |- ((_V e. Fin /\ A e. V /\ B e. W) -> Ncfin B e. Nn )
9	8	adantr 451	. . . 4 |- (((_V e. Fin /\ A e. V /\ B e. W) /\ A e. Ncfin B) -> Ncfin B e. Nn )
10	3	simprd 449	. . . . 5 |- ((_V e. Fin /\ A e. V /\ B e. W) -> A e. Ncfin A)
11	10	adantr 451	. . . 4 |- (((_V e. Fin /\ A e. V /\ B e. W) /\ A e. Ncfin B) -> A e. Ncfin A)
12		simpr 447	. . . 4 |- (((_V e. Fin /\ A e. V /\ B e. W) /\ A e. Ncfin B) -> A e. Ncfin B)
13		nnceleq 4430	. . . 4 |- ((( Ncfin A e. Nn /\ Ncfin B e. Nn ) /\ (A e. Ncfin A /\ A e. Ncfin B)) -> Ncfin A = Ncfin B)
14	5, 9, 11, 12, 13	syl22anc 1183	. . 3 |- (((_V e. Fin /\ A e. V /\ B e. W) /\ A e. Ncfin B) -> Ncfin A = Ncfin B)
15	14	ex 423	. 2 |- ((_V e. Fin /\ A e. V /\ B e. W) -> (A e. Ncfin B -> Ncfin A = Ncfin B))
16		eleq2 2414	. . 3 |- ( Ncfin A = Ncfin B -> (A e. Ncfin A <-> A e. Ncfin B))
17	10, 16	syl5ibcom 211	. 2 |- ((_V e. Fin /\ A e. V /\ B e. W) -> ( Ncfin A = Ncfin B -> A e. Ncfin B))
18	15, 17	impbid 183	1 |- ((_V e. Fin /\ A e. V /\ B e. W) -> (A e. Ncfin B <-> Ncfin A = Ncfin B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  _Vcvv 2859   Nn cnnc 4373   Fin cfin 4376   Nc_fin cncfin 4434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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)