Theorem spacssnc 6285

Description: The special set generator generates a set of cardinals. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
spacssnc	⊢ (N ∈ NC → ( Spac ‘N) ⊆ NC )

Proof of Theorem spacssnc

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

Step	Hyp	Ref	Expression
1		spacval 6283	. 2 ⊢ (N ∈ NC → ( Spac ‘N) = Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}))
2		snex 4112	. . . 4 ⊢ {N} ∈ V
3		spacvallem1 6282	. . . 4 ⊢ {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} ∈ V
4		eqid 2353	. . . 4 ⊢ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}) = Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})
5	2, 3, 4	clos1baseima 5884	. . 3 ⊢ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}) = ({N} ∪ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})))
6		snssi 3853	. . . . 5 ⊢ (N ∈ NC → {N} ⊆ NC )
7		imassrn 5010	. . . . . 6 ⊢ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})) ⊆ ran {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}
8		rnopab 4968	. . . . . . 7 ⊢ ran {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} = {y ∣ ∃x(x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}
9		simp2 956	. . . . . . . . 9 ⊢ ((x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x)) → y ∈ NC )
10	9	exlimiv 1634	. . . . . . . 8 ⊢ (∃x(x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x)) → y ∈ NC )
11	10	abssi 3342	. . . . . . 7 ⊢ {y ∣ ∃x(x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} ⊆ NC
12	8, 11	eqsstri 3302	. . . . . 6 ⊢ ran {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} ⊆ NC
13	7, 12	sstri 3282	. . . . 5 ⊢ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})) ⊆ NC
14	6, 13	jctir 524	. . . 4 ⊢ (N ∈ NC → ({N} ⊆ NC ∧ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})) ⊆ NC ))
15		unss 3438	. . . 4 ⊢ (({N} ⊆ NC ∧ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})) ⊆ NC ) ↔ ({N} ∪ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}))) ⊆ NC )
16	14, 15	sylib 188	. . 3 ⊢ (N ∈ NC → ({N} ∪ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}))) ⊆ NC )
17	5, 16	syl5eqss 3316	. 2 ⊢ (N ∈ NC → Clos1 ({N}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}) ⊆ NC )
18	1, 17	eqsstrd 3306	1 ⊢ (N ∈ NC → ( Spac ‘N) ⊆ NC )

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ∪ cun 3208   ⊆ wss 3258  {csn 3738  {copab 4623   “ cima 4723  ran crn 4774   ‘cfv 4782  (class class class)co 5526   Clos1 cclos1 5873   NC cncs 6089  2_cc2c 6095   ↑_c cce 6097   Sp_ac cspac 6274

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-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-2c 6105  df-ce 6107  df-spac 6275

This theorem is referenced by:  spaccl  6287  spacind  6288  nchoicelem4  6293  nchoicelem6  6295