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Theorem spacssnc 6284
 Description: The special set generator generates a set of cardinals. (Contributed by SF, 13-Mar-2015.)
Assertion
Ref Expression
spacssnc (N NC → ( SpacN) NC )

Proof of Theorem spacssnc
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 spacval 6282 . 2 (N NC → ( SpacN) = Clos1 ({N}, {x, y (x NC y NC y = (2cc x))}))
2 snex 4111 . . . 4 {N} V
3 spacvallem1 6281 . . . 4 {x, y (x NC y NC y = (2cc x))} V
4 eqid 2353 . . . 4 Clos1 ({N}, {x, y (x NC y NC y = (2cc x))}) = Clos1 ({N}, {x, y (x NC y NC y = (2cc x))})
52, 3, 4clos1baseima 5883 . . 3 Clos1 ({N}, {x, y (x NC y NC y = (2cc x))}) = ({N} ∪ ({x, y (x NC y NC y = (2cc x))} “ Clos1 ({N}, {x, y (x NC y NC y = (2cc x))})))
6 snssi 3852 . . . . 5 (N NC → {N} NC )
7 imassrn 5009 . . . . . 6 ({x, y (x NC y NC y = (2cc x))} “ Clos1 ({N}, {x, y (x NC y NC y = (2cc x))})) ran {x, y (x NC y NC y = (2cc x))}
8 rnopab 4967 . . . . . . 7 ran {x, y (x NC y NC y = (2cc x))} = {y x(x NC y NC y = (2cc x))}
9 simp2 956 . . . . . . . . 9 ((x NC y NC y = (2cc x)) → y NC )
109exlimiv 1634 . . . . . . . 8 (x(x NC y NC y = (2cc x)) → y NC )
1110abssi 3341 . . . . . . 7 {y x(x NC y NC y = (2cc x))} NC
128, 11eqsstri 3301 . . . . . 6 ran {x, y (x NC y NC y = (2cc x))} NC
137, 12sstri 3281 . . . . 5 ({x, y (x NC y NC y = (2cc x))} “ Clos1 ({N}, {x, y (x NC y NC y = (2cc x))})) NC
146, 13jctir 524 . . . 4 (N NC → ({N} NC ({x, y (x NC y NC y = (2cc x))} “ Clos1 ({N}, {x, y (x NC y NC y = (2cc x))})) NC ))
15 unss 3437 . . . 4 (({N} NC ({x, y (x NC y NC y = (2cc x))} “ Clos1 ({N}, {x, y (x NC y NC y = (2cc x))})) NC ) ↔ ({N} ∪ ({x, y (x NC y NC y = (2cc x))} “ Clos1 ({N}, {x, y (x NC y NC y = (2cc x))}))) NC )
1614, 15sylib 188 . . 3 (N NC → ({N} ∪ ({x, y (x NC y NC y = (2cc x))} “ Clos1 ({N}, {x, y (x NC y NC y = (2cc x))}))) NC )
175, 16syl5eqss 3315 . 2 (N NC Clos1 ({N}, {x, y (x NC y NC y = (2cc x))}) NC )
181, 17eqsstrd 3305 1 (N NC → ( SpacN) NC )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ∪ cun 3207   ⊆ wss 3257  {csn 3737  {copab 4622   “ cima 4722  ran crn 4773   ‘cfv 4781  (class class class)co 5525   Clos1 cclos1 5872   NC cncs 6088  2cc2c 6094   ↑c cce 6096   Spac cspac 6273 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-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-2c 6104  df-ce 6106  df-spac 6274 This theorem is referenced by:  spaccl  6286  spacind  6287  nchoicelem4  6292  nchoicelem6  6294
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