Theorem ncfindi 4476

Description: Distribution law for finite cardinality. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
ncfindi	⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → Ncfin (A ∪ B) = ( Ncfin A +c Ncfin B))

Proof of Theorem ncfindi

Step	Hyp	Ref	Expression
1		simp1l 979	. . . 4 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → V ∈ Fin )
2		simp1r 980	. . . . 5 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → A ∈ V)
3		simp2 956	. . . . 5 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → B ∈ W)
4		unexg 4102	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∪ B) ∈ V)
5	2, 3, 4	syl2anc 642	. . . 4 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → (A ∪ B) ∈ V)
6		ncfinprop 4475	. . . 4 ⊢ ((V ∈ Fin ∧ (A ∪ B) ∈ V) → ( Ncfin (A ∪ B) ∈ Nn ∧ (A ∪ B) ∈ Ncfin (A ∪ B)))
7	1, 5, 6	syl2anc 642	. . 3 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → ( Ncfin (A ∪ B) ∈ Nn ∧ (A ∪ B) ∈ Ncfin (A ∪ B)))
8	7	simpld 445	. 2 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → Ncfin (A ∪ B) ∈ Nn )
9		ncfinprop 4475	. . . . 5 ⊢ ((V ∈ Fin ∧ A ∈ V) → ( Ncfin A ∈ Nn ∧ A ∈ Ncfin A))
10	1, 2, 9	syl2anc 642	. . . 4 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → ( Ncfin A ∈ Nn ∧ A ∈ Ncfin A))
11	10	simpld 445	. . 3 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → Ncfin A ∈ Nn )
12		ncfinprop 4475	. . . . 5 ⊢ ((V ∈ Fin ∧ B ∈ W) → ( Ncfin B ∈ Nn ∧ B ∈ Ncfin B))
13	1, 3, 12	syl2anc 642	. . . 4 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → ( Ncfin B ∈ Nn ∧ B ∈ Ncfin B))
14	13	simpld 445	. . 3 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → Ncfin B ∈ Nn )
15		nncaddccl 4420	. . 3 ⊢ (( Ncfin A ∈ Nn ∧ Ncfin B ∈ Nn ) → ( Ncfin A +c Ncfin B) ∈ Nn )
16	11, 14, 15	syl2anc 642	. 2 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → ( Ncfin A +c Ncfin B) ∈ Nn )
17	7	simprd 449	. 2 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → (A ∪ B) ∈ Ncfin (A ∪ B))
18	10	simprd 449	. . 3 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → A ∈ Ncfin A)
19	13	simprd 449	. . 3 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → B ∈ Ncfin B)
20		simp3 957	. . 3 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → (A ∩ B) = ∅)
21		eladdci 4400	. . 3 ⊢ ((A ∈ Ncfin A ∧ B ∈ Ncfin B ∧ (A ∩ B) = ∅) → (A ∪ B) ∈ ( Ncfin A +c Ncfin B))
22	18, 19, 20, 21	syl3anc 1182	. 2 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → (A ∪ B) ∈ ( Ncfin A +c Ncfin B))
23		nnceleq 4431	. 2 ⊢ ((( Ncfin (A ∪ B) ∈ Nn ∧ ( Ncfin A +c Ncfin B) ∈ Nn ) ∧ ((A ∪ B) ∈ Ncfin (A ∪ B) ∧ (A ∪ B) ∈ ( Ncfin A +c Ncfin B))) → Ncfin (A ∪ B) = ( Ncfin A +c Ncfin B))
24	8, 16, 17, 22, 23	syl22anc 1183	1 ⊢ (((V ∈ Fin ∧ A ∈ V) ∧ B ∈ W ∧ (A ∩ B) = ∅) → Ncfin (A ∪ B) = ( Ncfin A +c Ncfin B))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208   ∩ cin 3209  ∅c0 3551   Nn cnnc 4374   +_c cplc 4376   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:  vfintle  4547  vfin1cltv  4548  vfinncsp  4555