Theorem ncaddccl 6145

Description: The cardinals are closed under cardinal addition. Theorem XI.2.10 of [Rosser] p. 374. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
ncaddccl	⊢ ((A ∈ NC ∧ B ∈ NC ) → (A +c B) ∈ NC )

Proof of Theorem ncaddccl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elncs 6120	. 2 ⊢ (A ∈ NC ↔ ∃x A = Nc x)
2		elncs 6120	. 2 ⊢ (B ∈ NC ↔ ∃y B = Nc y)
3		eeanv 1913	. . 3 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) ↔ (∃x A = Nc x ∧ ∃y B = Nc y))
4		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
5		0ex 4111	. . . . . . . . . 10 ⊢ ∅ ∈ V
6	5	complex 4105	. . . . . . . . 9 ⊢ ∼ ∅ ∈ V
7	4, 6	xpsnen 6050	. . . . . . . 8 ⊢ (x × { ∼ ∅}) ≈ x
8		snex 4112	. . . . . . . . . 10 ⊢ { ∼ ∅} ∈ V
9	4, 8	xpex 5116	. . . . . . . . 9 ⊢ (x × { ∼ ∅}) ∈ V
10	9	eqnc 6128	. . . . . . . 8 ⊢ ( Nc (x × { ∼ ∅}) = Nc x ↔ (x × { ∼ ∅}) ≈ x)
11	7, 10	mpbir 200	. . . . . . 7 ⊢ Nc (x × { ∼ ∅}) = Nc x
12	11	eqcomi 2357	. . . . . 6 ⊢ Nc x = Nc (x × { ∼ ∅})
13		eqtr 2370	. . . . . 6 ⊢ ((A = Nc x ∧ Nc x = Nc (x × { ∼ ∅})) → A = Nc (x × { ∼ ∅}))
14	12, 13	mpan2 652	. . . . 5 ⊢ (A = Nc x → A = Nc (x × { ∼ ∅}))
15		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
16	15, 5	xpsnen 6050	. . . . . . . 8 ⊢ (y × {∅}) ≈ y
17		snex 4112	. . . . . . . . . 10 ⊢ {∅} ∈ V
18	15, 17	xpex 5116	. . . . . . . . 9 ⊢ (y × {∅}) ∈ V
19	18	eqnc 6128	. . . . . . . 8 ⊢ ( Nc (y × {∅}) = Nc y ↔ (y × {∅}) ≈ y)
20	16, 19	mpbir 200	. . . . . . 7 ⊢ Nc (y × {∅}) = Nc y
21	20	eqcomi 2357	. . . . . 6 ⊢ Nc y = Nc (y × {∅})
22		eqtr 2370	. . . . . 6 ⊢ ((B = Nc y ∧ Nc y = Nc (y × {∅})) → B = Nc (y × {∅}))
23	21, 22	mpan2 652	. . . . 5 ⊢ (B = Nc y → B = Nc (y × {∅}))
24		addceq12 4386	. . . . . 6 ⊢ ((A = Nc (x × { ∼ ∅}) ∧ B = Nc (y × {∅})) → (A +c B) = ( Nc (x × { ∼ ∅}) +c Nc (y × {∅})))
25		necompl 3545	. . . . . . . . . . 11 ⊢ ∼ ∅ ≠ ∅
26	6, 25	xpnedisj 5514	. . . . . . . . . 10 ⊢ ((x × { ∼ ∅}) ∩ (y × {∅})) = ∅
27	9, 18	ncdisjun 6137	. . . . . . . . . 10 ⊢ (((x × { ∼ ∅}) ∩ (y × {∅})) = ∅ → Nc ((x × { ∼ ∅}) ∪ (y × {∅})) = ( Nc (x × { ∼ ∅}) +c Nc (y × {∅})))
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ Nc ((x × { ∼ ∅}) ∪ (y × {∅})) = ( Nc (x × { ∼ ∅}) +c Nc (y × {∅}))
29	28	eqcomi 2357	. . . . . . . 8 ⊢ ( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) = Nc ((x × { ∼ ∅}) ∪ (y × {∅}))
30	9, 18	unex 4107	. . . . . . . . 9 ⊢ ((x × { ∼ ∅}) ∪ (y × {∅})) ∈ V
31		nceq 6109	. . . . . . . . . 10 ⊢ (z = ((x × { ∼ ∅}) ∪ (y × {∅})) → Nc z = Nc ((x × { ∼ ∅}) ∪ (y × {∅})))
32	31	eqeq2d 2364	. . . . . . . . 9 ⊢ (z = ((x × { ∼ ∅}) ∪ (y × {∅})) → (( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) = Nc z ↔ ( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) = Nc ((x × { ∼ ∅}) ∪ (y × {∅}))))
33	30, 32	spcev 2947	. . . . . . . 8 ⊢ (( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) = Nc ((x × { ∼ ∅}) ∪ (y × {∅})) → ∃z( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) = Nc z)
34	29, 33	ax-mp 5	. . . . . . 7 ⊢ ∃z( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) = Nc z
35		elncs 6120	. . . . . . 7 ⊢ (( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) ∈ NC ↔ ∃z( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) = Nc z)
36	34, 35	mpbir 200	. . . . . 6 ⊢ ( Nc (x × { ∼ ∅}) +c Nc (y × {∅})) ∈ NC
37	24, 36	syl6eqel 2441	. . . . 5 ⊢ ((A = Nc (x × { ∼ ∅}) ∧ B = Nc (y × {∅})) → (A +c B) ∈ NC )
38	14, 23, 37	syl2an 463	. . . 4 ⊢ ((A = Nc x ∧ B = Nc y) → (A +c B) ∈ NC )
39	38	exlimivv 1635	. . 3 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) → (A +c B) ∈ NC )
40	3, 39	sylbir 204	. 2 ⊢ ((∃x A = Nc x ∧ ∃y B = Nc y) → (A +c B) ∈ NC )
41	1, 2, 40	syl2anb 465	1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A +c B) ∈ NC )

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∼ ccompl 3206   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738   +_c cplc 4376   class class class wbr 4640   × cxp 4771   ≈ cen 6029   NC cncs 6089   Nc cnc 6092

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-13 1712  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-3or 935  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-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102

This theorem is referenced by:  peano2nc  6146  tcdi  6165  ce0addcnnul  6180  addlecncs  6210  lectr  6212  leaddc1  6215  taddc  6230  tlecg  6231  letc  6232  nclenn  6250  addcdi  6251  addcdir  6252  addccan2nc  6266  lecadd2  6267  ncslesuc  6268  nchoicelem1  6290