Theorem nchoicelem5 6294

Description: Lemma for nchoice 6309. A cardinal is not a member of the special set of itself raised to two. Theorem 6.5 of [Specker] p. 973. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
nchoicelem5	⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ¬ M ∈ ( Spac ‘(2c ↑c M)))

Proof of Theorem nchoicelem5

Step	Hyp	Ref	Expression
1		ce2lt 6221	. . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → M <c (2c ↑c M))
2		2nc 6169	. . . . . . 7 ⊢ 2c ∈ NC
3	2	jctl 525	. . . . . 6 ⊢ (M ∈ NC → (2c ∈ NC ∧ M ∈ NC ))
4		2nnc 6168	. . . . . . . . 9 ⊢ 2c ∈ Nn
5		ce0nn 6181	. . . . . . . . 9 ⊢ (2c ∈ Nn → (2c ↑c 0c) ≠ ∅)
6	4, 5	ax-mp 5	. . . . . . . 8 ⊢ (2c ↑c 0c) ≠ ∅
7		ce0nulnc 6185	. . . . . . . . 9 ⊢ (2c ∈ NC → ((2c ↑c 0c) ≠ ∅ ↔ (2c ↑c 0c) ∈ NC ))
8	2, 7	ax-mp 5	. . . . . . . 8 ⊢ ((2c ↑c 0c) ≠ ∅ ↔ (2c ↑c 0c) ∈ NC )
9	6, 8	mpbi 199	. . . . . . 7 ⊢ (2c ↑c 0c) ∈ NC
10	9	jctl 525	. . . . . 6 ⊢ ((M ↑c 0c) ∈ NC → ((2c ↑c 0c) ∈ NC ∧ (M ↑c 0c) ∈ NC ))
11		cecl 6187	. . . . . 6 ⊢ (((2c ∈ NC ∧ M ∈ NC ) ∧ ((2c ↑c 0c) ∈ NC ∧ (M ↑c 0c) ∈ NC )) → (2c ↑c M) ∈ NC )
12	3, 10, 11	syl2an 463	. . . . 5 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ NC )
13		ltlenlec 6208	. . . . 5 ⊢ ((M ∈ NC ∧ (2c ↑c M) ∈ NC ) → (M <c (2c ↑c M) ↔ (M ≤c (2c ↑c M) ∧ ¬ (2c ↑c M) ≤c M)))
14	12, 13	syldan 456	. . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (M <c (2c ↑c M) ↔ (M ≤c (2c ↑c M) ∧ ¬ (2c ↑c M) ≤c M)))
15	1, 14	mpbid 201	. . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (M ≤c (2c ↑c M) ∧ ¬ (2c ↑c M) ≤c M))
16	15	simprd 449	. 2 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ¬ (2c ↑c M) ≤c M)
17		nchoicelem4 6293	. . 3 ⊢ (((2c ↑c M) ∈ NC ∧ M ∈ ( Spac ‘(2c ↑c M))) → (2c ↑c M) ≤c M)
18	12, 17	sylan 457	. 2 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ M ∈ ( Spac ‘(2c ↑c M))) → (2c ↑c M) ≤c M)
19	16, 18	mtand 640	1 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ¬ M ∈ ( Spac ‘(2c ↑c M)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2517  ∅c0 3551   Nn cnnc 4374  0_cc0c 4375   class class class wbr 4640   ‘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:  nchoicelem7  6296  nchoicelem14  6303