Theorem nchoicelem4 6293

Description: Lemma for nchoice 6309. The initial value of Sp_ac is a minimum value. Theorem 6.4 of [Specker] p. 973. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
nchoicelem4	⊢ ((M ∈ NC ∧ N ∈ ( Spac ‘M)) → M ≤c N)

Proof of Theorem nchoicelem4

Dummy variables n p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasn 5019	. . 3 ⊢ ( ≤c “ {M}) = {p ∣ M ≤c p}
2		lecex 6116	. . . 4 ⊢ ≤c ∈ V
3		snex 4112	. . . 4 ⊢ {M} ∈ V
4	2, 3	imaex 4748	. . 3 ⊢ ( ≤c “ {M}) ∈ V
5	1, 4	eqeltrri 2424	. 2 ⊢ {p ∣ M ≤c p} ∈ V
6		breq2 4644	. 2 ⊢ (p = M → (M ≤c p ↔ M ≤c M))
7		breq2 4644	. 2 ⊢ (p = n → (M ≤c p ↔ M ≤c n))
8		breq2 4644	. 2 ⊢ (p = (2c ↑c n) → (M ≤c p ↔ M ≤c (2c ↑c n)))
9		breq2 4644	. 2 ⊢ (p = N → (M ≤c p ↔ M ≤c N))
10		nclecid 6198	. 2 ⊢ (M ∈ NC → M ≤c M)
11		spacssnc 6285	. . . . . . 7 ⊢ (M ∈ NC → ( Spac ‘M) ⊆ NC )
12	11	sselda 3274	. . . . . 6 ⊢ ((M ∈ NC ∧ n ∈ ( Spac ‘M)) → n ∈ NC )
13		ce2lt 6221	. . . . . 6 ⊢ ((n ∈ NC ∧ (n ↑c 0c) ∈ NC ) → n <c (2c ↑c n))
14	12, 13	sylan 457	. . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → n <c (2c ↑c n))
15		brltc 6115	. . . . . 6 ⊢ (n <c (2c ↑c n) ↔ (n ≤c (2c ↑c n) ∧ n ≠ (2c ↑c n)))
16	15	simplbi 446	. . . . 5 ⊢ (n <c (2c ↑c n) → n ≤c (2c ↑c n))
17	14, 16	syl 15	. . . 4 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → n ≤c (2c ↑c n))
18		simpll 730	. . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → M ∈ NC )
19	12	adantr 451	. . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → n ∈ NC )
20		2nnc 6168	. . . . . . 7 ⊢ 2c ∈ Nn
21		ceclnn1 6190	. . . . . . 7 ⊢ ((2c ∈ Nn ∧ n ∈ NC ∧ (n ↑c 0c) ∈ NC ) → (2c ↑c n) ∈ NC )
22	20, 21	mp3an1 1264	. . . . . 6 ⊢ ((n ∈ NC ∧ (n ↑c 0c) ∈ NC ) → (2c ↑c n) ∈ NC )
23	12, 22	sylan 457	. . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → (2c ↑c n) ∈ NC )
24		lectr 6212	. . . . 5 ⊢ ((M ∈ NC ∧ n ∈ NC ∧ (2c ↑c n) ∈ NC ) → ((M ≤c n ∧ n ≤c (2c ↑c n)) → M ≤c (2c ↑c n)))
25	18, 19, 23, 24	syl3anc 1182	. . . 4 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → ((M ≤c n ∧ n ≤c (2c ↑c n)) → M ≤c (2c ↑c n)))
26	17, 25	mpan2d 655	. . 3 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → (M ≤c n → M ≤c (2c ↑c n)))
27	26	impr 602	. 2 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ ((n ↑c 0c) ∈ NC ∧ M ≤c n)) → M ≤c (2c ↑c n))
28	5, 6, 7, 8, 9, 10, 27	spacis 6289	1 ⊢ ((M ∈ NC ∧ N ∈ ( Spac ‘M)) → M ≤c N)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  {cab 2339   ≠ wne 2517  Vcvv 2860  {csn 3738   Nn cnnc 4374  0_cc0c 4375   class class class wbr 4640   “ cima 4723   ‘cfv 4782  (class class class)co 5526   NC cncs 6089   ≤_c clec 6090   <_c cltc 6091  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-compose 5749  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-lec 6100  df-ltc 6101  df-nc 6102  df-2c 6105  df-ce 6107  df-spac 6275

This theorem is referenced by:  nchoicelem5  6294