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 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 )

Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . 2 (A NCx A = Nc x)
2 elncs 6119 . 2 (B NCy B = Nc y)
3 eeanv 1913 . . 3 (xy(A = Nc x B = Nc y) ↔ (x A = Nc x y B = Nc y))
4 vex 2862 . . . . . . . . 9 x V
5 0ex 4110 . . . . . . . . . 10 V
65complex 4104 . . . . . . . . 9 V
74, 6xpsnen 6049 . . . . . . . 8 (x × { ∼ }) ≈ x
8 snex 4111 . . . . . . . . . 10 { ∼ } V
94, 8xpex 5115 . . . . . . . . 9 (x × { ∼ }) V
109eqnc 6127 . . . . . . . 8 ( Nc (x × { ∼ }) = Nc x ↔ (x × { ∼ }) ≈ x)
117, 10mpbir 200 . . . . . . 7 Nc (x × { ∼ }) = Nc x
1211eqcomi 2357 . . . . . 6 Nc x = Nc (x × { ∼ })
13 eqtr 2370 . . . . . 6 ((A = Nc x Nc x = Nc (x × { ∼ })) → A = Nc (x × { ∼ }))
1412, 13mpan2 652 . . . . 5 (A = Nc xA = Nc (x × { ∼ }))
15 vex 2862 . . . . . . . . 9 y V
1615, 5xpsnen 6049 . . . . . . . 8 (y × {}) ≈ y
17 snex 4111 . . . . . . . . . 10 {} V
1815, 17xpex 5115 . . . . . . . . 9 (y × {}) V
1918eqnc 6127 . . . . . . . 8 ( Nc (y × {}) = Nc y ↔ (y × {}) ≈ y)
2016, 19mpbir 200 . . . . . . 7 Nc (y × {}) = Nc y
2120eqcomi 2357 . . . . . 6 Nc y = Nc (y × {})
22 eqtr 2370 . . . . . 6 ((B = Nc y Nc y = Nc (y × {})) → B = Nc (y × {}))
2321, 22mpan2 652 . . . . 5 (B = Nc yB = Nc (y × {}))
24 addceq12 4385 . . . . . 6 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → (A +c B) = ( Nc (x × { ∼ }) +c Nc (y × {})))
25 necompl 3544 . . . . . . . . . . 11
266, 25xpnedisj 5513 . . . . . . . . . 10 ((x × { ∼ }) ∩ (y × {})) =
279, 18ncdisjun 6136 . . . . . . . . . 10 (((x × { ∼ }) ∩ (y × {})) = Nc ((x × { ∼ }) ∪ (y × {})) = ( Nc (x × { ∼ }) +c Nc (y × {})))
2826, 27ax-mp 8 . . . . . . . . 9 Nc ((x × { ∼ }) ∪ (y × {})) = ( Nc (x × { ∼ }) +c Nc (y × {}))
2928eqcomi 2357 . . . . . . . 8 ( Nc (x × { ∼ }) +c Nc (y × {})) = Nc ((x × { ∼ }) ∪ (y × {}))
309, 18unex 4106 . . . . . . . . 9 ((x × { ∼ }) ∪ (y × {})) V
31 nceq 6108 . . . . . . . . . 10 (z = ((x × { ∼ }) ∪ (y × {})) → Nc z = Nc ((x × { ∼ }) ∪ (y × {})))
3231eqeq2d 2364 . . . . . . . . 9 (z = ((x × { ∼ }) ∪ (y × {})) → (( Nc (x × { ∼ }) +c Nc (y × {})) = Nc z ↔ ( Nc (x × { ∼ }) +c Nc (y × {})) = Nc ((x × { ∼ }) ∪ (y × {}))))
3330, 32spcev 2946 . . . . . . . 8 (( Nc (x × { ∼ }) +c Nc (y × {})) = Nc ((x × { ∼ }) ∪ (y × {})) → z( Nc (x × { ∼ }) +c Nc (y × {})) = Nc z)
3429, 33ax-mp 8 . . . . . . 7 z( Nc (x × { ∼ }) +c Nc (y × {})) = Nc z
35 elncs 6119 . . . . . . 7 (( Nc (x × { ∼ }) +c Nc (y × {})) NCz( Nc (x × { ∼ }) +c Nc (y × {})) = Nc z)
3634, 35mpbir 200 . . . . . 6 ( Nc (x × { ∼ }) +c Nc (y × {})) NC
3724, 36syl6eqel 2441 . . . . 5 ((A = Nc (x × { ∼ }) B = Nc (y × {})) → (A +c B) NC )
3814, 23, 37syl2an 463 . . . 4 ((A = Nc x B = Nc y) → (A +c B) NC )
3938exlimivv 1635 . . 3 (xy(A = Nc x B = Nc y) → (A +c B) NC )
403, 39sylbir 204 . 2 ((x A = Nc x y B = Nc y) → (A +c B) NC )
411, 2, 40syl2anb 465 1 ((A NC B NC ) → (A +c B) NC )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737   +c cplc 4375   class class class wbr 4639   × cxp 4770   ≈ cen 6028   NC cncs 6088   Nc cnc 6091 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 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-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 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-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101 This theorem is referenced by:  peano2nc  6145  tcdi  6164  ce0addcnnul  6179  addlecncs  6209  lectr  6211  leaddc1  6214  taddc  6229  tlecg  6230  letc  6231  nclenn  6249  addcdi  6250  addcdir  6251  addccan2nc  6265  lecadd2  6266  ncslesuc  6267  nchoicelem1  6289
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