Theorem ncfindi 4475

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 4101	. . . . 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 4474	. . . 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 4474	. . . . 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 4474	. . . . 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 4419	. . 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 4399	. . 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 4430	. 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 2859   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   Nn cnnc 4373   +_c cplc 4375   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:  vfintle  4546  vfin1cltv  4547  vfinncsp  4554