Theorem nclennlem1 6249

Description: Lemma for nclenn 6250. Set up stratification for induction. (Contributed by SF, 19-Mar-2015.)

Assertion

Ref	Expression
nclennlem1	⊢ {x ∣ ∀n ∈ NC (n ≤c x → n ∈ Nn )} ∈ V

Distinct variable group:   x,n

Proof of Theorem nclennlem1

Step	Hyp	Ref	Expression
1		vex 2863	. . . . 5 ⊢ x ∈ V
2	1	elcompl 3226	. . . 4 ⊢ (x ∈ ∼ (( ≤c ↾ ∼ Nn ) “ NC ) ↔ ¬ x ∈ (( ≤c ↾ ∼ Nn ) “ NC ))
3		elima 4755	. . . . . 6 ⊢ (x ∈ (( ≤c ↾ ∼ Nn ) “ NC ) ↔ ∃n ∈ NC n( ≤c ↾ ∼ Nn )x)
4		brres 4950	. . . . . . . 8 ⊢ (n( ≤c ↾ ∼ Nn )x ↔ (n ≤c x ∧ n ∈ ∼ Nn ))
5		vex 2863	. . . . . . . . . 10 ⊢ n ∈ V
6	5	elcompl 3226	. . . . . . . . 9 ⊢ (n ∈ ∼ Nn ↔ ¬ n ∈ Nn )
7	6	anbi2i 675	. . . . . . . 8 ⊢ ((n ≤c x ∧ n ∈ ∼ Nn ) ↔ (n ≤c x ∧ ¬ n ∈ Nn ))
8	4, 7	bitri 240	. . . . . . 7 ⊢ (n( ≤c ↾ ∼ Nn )x ↔ (n ≤c x ∧ ¬ n ∈ Nn ))
9	8	rexbii 2640	. . . . . 6 ⊢ (∃n ∈ NC n( ≤c ↾ ∼ Nn )x ↔ ∃n ∈ NC (n ≤c x ∧ ¬ n ∈ Nn ))
10		rexanali 2661	. . . . . 6 ⊢ (∃n ∈ NC (n ≤c x ∧ ¬ n ∈ Nn ) ↔ ¬ ∀n ∈ NC (n ≤c x → n ∈ Nn ))
11	3, 9, 10	3bitrri 263	. . . . 5 ⊢ (¬ ∀n ∈ NC (n ≤c x → n ∈ Nn ) ↔ x ∈ (( ≤c ↾ ∼ Nn ) “ NC ))
12	11	con1bii 321	. . . 4 ⊢ (¬ x ∈ (( ≤c ↾ ∼ Nn ) “ NC ) ↔ ∀n ∈ NC (n ≤c x → n ∈ Nn ))
13	2, 12	bitri 240	. . 3 ⊢ (x ∈ ∼ (( ≤c ↾ ∼ Nn ) “ NC ) ↔ ∀n ∈ NC (n ≤c x → n ∈ Nn ))
14	13	abbi2i 2465	. 2 ⊢ ∼ (( ≤c ↾ ∼ Nn ) “ NC ) = {x ∣ ∀n ∈ NC (n ≤c x → n ∈ Nn )}
15		lecex 6116	. . . . 5 ⊢ ≤c ∈ V
16		nncex 4397	. . . . . 6 ⊢ Nn ∈ V
17	16	complex 4105	. . . . 5 ⊢ ∼ Nn ∈ V
18	15, 17	resex 5118	. . . 4 ⊢ ( ≤c ↾ ∼ Nn ) ∈ V
19		ncsex 6112	. . . 4 ⊢ NC ∈ V
20	18, 19	imaex 4748	. . 3 ⊢ (( ≤c ↾ ∼ Nn ) “ NC ) ∈ V
21	20	complex 4105	. 2 ⊢ ∼ (( ≤c ↾ ∼ Nn ) “ NC ) ∈ V
22	14, 21	eqeltrri 2424	1 ⊢ {x ∣ ∀n ∈ NC (n ≤c x → n ∈ Nn )} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   Nn cnnc 4374   class class class wbr 4640   “ cima 4723   ↾ cres 4775   NC cncs 6089   ≤_c clec 6090

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-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100

This theorem is referenced by:  nclenn  6250