Theorem nchoicelem14 6303

Description: Lemma for nchoice 6309. When the special set generator yields a singleton, then the cardinal is not raisable. (Contributed by SF, 19-Mar-2015.)

Assertion

Ref	Expression
nchoicelem14	⊢ ((M ∈ NC ∧ Nc ( Spac ‘M) = 1c) → ¬ (M ↑c 0c) ∈ NC )

Proof of Theorem nchoicelem14

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nchoicelem5 6294	. . . . . . . . 9 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ¬ M ∈ ( Spac ‘(2c ↑c M)))
2		incom 3449	. . . . . . . . . . 11 ⊢ ({M} ∩ ( Spac ‘(2c ↑c M))) = (( Spac ‘(2c ↑c M)) ∩ {M})
3	2	eqeq1i 2360	. . . . . . . . . 10 ⊢ (({M} ∩ ( Spac ‘(2c ↑c M))) = ∅ ↔ (( Spac ‘(2c ↑c M)) ∩ {M}) = ∅)
4		disjsn 3787	. . . . . . . . . 10 ⊢ ((( Spac ‘(2c ↑c M)) ∩ {M}) = ∅ ↔ ¬ M ∈ ( Spac ‘(2c ↑c M)))
5	3, 4	bitri 240	. . . . . . . . 9 ⊢ (({M} ∩ ( Spac ‘(2c ↑c M))) = ∅ ↔ ¬ M ∈ ( Spac ‘(2c ↑c M)))
6	1, 5	sylibr 203	. . . . . . . 8 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ({M} ∩ ( Spac ‘(2c ↑c M))) = ∅)
7		snex 4112	. . . . . . . . 9 ⊢ {M} ∈ V
8		fvex 5340	. . . . . . . . 9 ⊢ ( Spac ‘(2c ↑c M)) ∈ V
9	7, 8	ncdisjun 6137	. . . . . . . 8 ⊢ (({M} ∩ ( Spac ‘(2c ↑c M))) = ∅ → Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = ( Nc {M} +c Nc ( Spac ‘(2c ↑c M))))
10	6, 9	syl 15	. . . . . . 7 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = ( Nc {M} +c Nc ( Spac ‘(2c ↑c M))))
11		df1c3g 6142	. . . . . . . . . . 11 ⊢ (M ∈ NC → 1c = Nc {M})
12	11	adantr 451	. . . . . . . . . 10 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → 1c = Nc {M})
13	12	addceq2d 4391	. . . . . . . . 9 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘(2c ↑c M)) +c 1c) = ( Nc ( Spac ‘(2c ↑c M)) +c Nc {M}))
14		addccom 4407	. . . . . . . . 9 ⊢ ( Nc {M} +c Nc ( Spac ‘(2c ↑c M))) = ( Nc ( Spac ‘(2c ↑c M)) +c Nc {M})
15	13, 14	syl6reqr 2404	. . . . . . . 8 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Nc {M} +c Nc ( Spac ‘(2c ↑c M))) = ( Nc ( Spac ‘(2c ↑c M)) +c 1c))
16		2nnc 6168	. . . . . . . . . . 11 ⊢ 2c ∈ Nn
17		ceclnn1 6190	. . . . . . . . . . 11 ⊢ ((2c ∈ Nn ∧ M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ NC )
18	16, 17	mp3an1 1264	. . . . . . . . . 10 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ NC )
19		nchoicelem13 6302	. . . . . . . . . 10 ⊢ ((2c ↑c M) ∈ NC → 1c ≤c Nc ( Spac ‘(2c ↑c M)))
20		1cnc 6140	. . . . . . . . . . . 12 ⊢ 1c ∈ NC
21	8	ncelncsi 6122	. . . . . . . . . . . 12 ⊢ Nc ( Spac ‘(2c ↑c M)) ∈ NC
22		dflec2 6211	. . . . . . . . . . . 12 ⊢ ((1c ∈ NC ∧ Nc ( Spac ‘(2c ↑c M)) ∈ NC ) → (1c ≤c Nc ( Spac ‘(2c ↑c M)) ↔ ∃k ∈ NC Nc ( Spac ‘(2c ↑c M)) = (1c +c k)))
23	20, 21, 22	mp2an 653	. . . . . . . . . . 11 ⊢ (1c ≤c Nc ( Spac ‘(2c ↑c M)) ↔ ∃k ∈ NC Nc ( Spac ‘(2c ↑c M)) = (1c +c k))
24		addccom 4407	. . . . . . . . . . . . . 14 ⊢ (1c +c k) = (k +c 1c)
25		0cnsuc 4402	. . . . . . . . . . . . . 14 ⊢ (k +c 1c) ≠ 0c
26	24, 25	eqnetri 2534	. . . . . . . . . . . . 13 ⊢ (1c +c k) ≠ 0c
27		neeq1 2525	. . . . . . . . . . . . 13 ⊢ ( Nc ( Spac ‘(2c ↑c M)) = (1c +c k) → ( Nc ( Spac ‘(2c ↑c M)) ≠ 0c ↔ (1c +c k) ≠ 0c))
28	26, 27	mpbiri 224	. . . . . . . . . . . 12 ⊢ ( Nc ( Spac ‘(2c ↑c M)) = (1c +c k) → Nc ( Spac ‘(2c ↑c M)) ≠ 0c)
29	28	rexlimivw 2735	. . . . . . . . . . 11 ⊢ (∃k ∈ NC Nc ( Spac ‘(2c ↑c M)) = (1c +c k) → Nc ( Spac ‘(2c ↑c M)) ≠ 0c)
30	23, 29	sylbi 187	. . . . . . . . . 10 ⊢ (1c ≤c Nc ( Spac ‘(2c ↑c M)) → Nc ( Spac ‘(2c ↑c M)) ≠ 0c)
31	18, 19, 30	3syl 18	. . . . . . . . 9 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ( Spac ‘(2c ↑c M)) ≠ 0c)
32		0cnc 6139	. . . . . . . . . . 11 ⊢ 0c ∈ NC
33		peano4nc 6151	. . . . . . . . . . 11 ⊢ (( Nc ( Spac ‘(2c ↑c M)) ∈ NC ∧ 0c ∈ NC ) → (( Nc ( Spac ‘(2c ↑c M)) +c 1c) = (0c +c 1c) ↔ Nc ( Spac ‘(2c ↑c M)) = 0c))
34	21, 32, 33	mp2an 653	. . . . . . . . . 10 ⊢ (( Nc ( Spac ‘(2c ↑c M)) +c 1c) = (0c +c 1c) ↔ Nc ( Spac ‘(2c ↑c M)) = 0c)
35	34	necon3bii 2549	. . . . . . . . 9 ⊢ (( Nc ( Spac ‘(2c ↑c M)) +c 1c) ≠ (0c +c 1c) ↔ Nc ( Spac ‘(2c ↑c M)) ≠ 0c)
36	31, 35	sylibr 203	. . . . . . . 8 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘(2c ↑c M)) +c 1c) ≠ (0c +c 1c))
37	15, 36	eqnetrd 2535	. . . . . . 7 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Nc {M} +c Nc ( Spac ‘(2c ↑c M))) ≠ (0c +c 1c))
38	10, 37	eqnetrd 2535	. . . . . 6 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ({M} ∪ ( Spac ‘(2c ↑c M))) ≠ (0c +c 1c))
39		addcid2 4408	. . . . . . . 8 ⊢ (0c +c 1c) = 1c
40	39	neeq2i 2528	. . . . . . 7 ⊢ ( Nc ({M} ∪ ( Spac ‘(2c ↑c M))) ≠ (0c +c 1c) ↔ Nc ({M} ∪ ( Spac ‘(2c ↑c M))) ≠ 1c)
41		df-ne 2519	. . . . . . 7 ⊢ ( Nc ({M} ∪ ( Spac ‘(2c ↑c M))) ≠ 1c ↔ ¬ Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = 1c)
42	40, 41	bitri 240	. . . . . 6 ⊢ ( Nc ({M} ∪ ( Spac ‘(2c ↑c M))) ≠ (0c +c 1c) ↔ ¬ Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = 1c)
43	38, 42	sylib 188	. . . . 5 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ¬ Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = 1c)
44		nchoicelem6 6295	. . . . . . 7 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) = ({M} ∪ ( Spac ‘(2c ↑c M))))
45	44	nceqd 6111	. . . . . 6 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ( Spac ‘M) = Nc ({M} ∪ ( Spac ‘(2c ↑c M))))
46	45	eqeq1d 2361	. . . . 5 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘M) = 1c ↔ Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = 1c))
47	43, 46	mtbird 292	. . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ¬ Nc ( Spac ‘M) = 1c)
48	47	ex 423	. . 3 ⊢ (M ∈ NC → ((M ↑c 0c) ∈ NC → ¬ Nc ( Spac ‘M) = 1c))
49	48	con2d 107	. 2 ⊢ (M ∈ NC → ( Nc ( Spac ‘M) = 1c → ¬ (M ↑c 0c) ∈ NC ))
50	49	imp 418	1 ⊢ ((M ∈ NC ∧ Nc ( Spac ‘M) = 1c) → ¬ (M ↑c 0c) ∈ NC )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   class class class wbr 4640   ‘cfv 4782  (class class class)co 5526   NC cncs 6089   ≤_c clec 6090   Nc cnc 6092  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:  nchoicelem17  6306