Theorem ncfineleq 4478

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 B ↔ Ncfin A = Ncfin B))

Proof of Theorem ncfineleq

Step	Hyp	Ref	Expression
1		simpl 443	. . . . 5 ⊢ (((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) ∧ A ∈ Ncfin B) → (V ∈ Fin ∧ A ∈ V ∧ B ∈ W))
2		ncfinprop 4475	. . . . . 6 ⊢ ((V ∈ Fin ∧ A ∈ V) → ( Ncfin A ∈ Nn ∧ A ∈ Ncfin A))
3	2	3adant3 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 )
5	1, 3, 4	3syl 18	. . . 4 ⊢ (((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) ∧ A ∈ Ncfin B) → Ncfin A ∈ Nn )
6		ncfinprop 4475	. . . . . . 7 ⊢ ((V ∈ Fin ∧ B ∈ W) → ( Ncfin B ∈ Nn ∧ B ∈ Ncfin B))
7	6	3adant2 974	. . . . . 6 ⊢ ((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) → ( Ncfin B ∈ Nn ∧ B ∈ Ncfin B))
8	7	simpld 445	. . . . 5 ⊢ ((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) → Ncfin B ∈ Nn )
9	8	adantr 451	. . . 4 ⊢ (((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) ∧ A ∈ Ncfin B) → Ncfin B ∈ Nn )
10	3	simprd 449	. . . . 5 ⊢ ((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) → A ∈ Ncfin A)
11	10	adantr 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 4431	. . . 4 ⊢ ((( Ncfin A ∈ Nn ∧ Ncfin B ∈ Nn ) ∧ (A ∈ Ncfin A ∧ A ∈ Ncfin B)) → Ncfin A = Ncfin B)
14	5, 9, 11, 12, 13	syl22anc 1183	. . 3 ⊢ (((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) ∧ A ∈ Ncfin B) → Ncfin A = Ncfin B)
15	14	ex 423	. 2 ⊢ ((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) → (A ∈ Ncfin B → Ncfin A = Ncfin B))
16		eleq2 2414	. . 3 ⊢ ( Ncfin A = Ncfin B → (A ∈ Ncfin A ↔ A ∈ Ncfin B))
17	10, 16	syl5ibcom 211	. 2 ⊢ ((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) → ( Ncfin A = Ncfin B → A ∈ Ncfin B))
18	15, 17	impbid 183	1 ⊢ ((V ∈ Fin ∧ A ∈ V ∧ B ∈ W) → (A ∈ Ncfin B ↔ Ncfin A = Ncfin B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2860   Nn cnnc 4374   Fin cfin 4377   Nc_fin cncfin 4435

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-ncfin 4443

This theorem is referenced by: (None)