Theorem ce0lenc1 6239

Description: Cardinal exponentiation to zero is a cardinal iff the number is less than the size of cardinal one. (Contributed by SF, 18-Mar-2015.)

Assertion

Ref	Expression
ce0lenc1	⊢ (M ∈ NC → ((M ↑c 0c) ∈ NC ↔ M ≤c Nc 1c))

Proof of Theorem ce0lenc1

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

Step	Hyp	Ref	Expression
1		ce0tb 6238	. 2 ⊢ (M ∈ NC → ((M ↑c 0c) ∈ NC ↔ ∃n ∈ NC M = Tc n))
2		elncs 6119	. . . . . 6 ⊢ (n ∈ NC ↔ ∃x n = Nc x)
3		tceq 6158	. . . . . . . . 9 ⊢ (n = Nc x → Tc n = Tc Nc x)
4		vex 2862	. . . . . . . . . 10 ⊢ x ∈ V
5	4	tcnc 6225	. . . . . . . . 9 ⊢ Tc Nc x = Nc ℘1x
6	3, 5	syl6eq 2401	. . . . . . . 8 ⊢ (n = Nc x → Tc n = Nc ℘1x)
7		pw1ss1c 4158	. . . . . . . . 9 ⊢ ℘1x ⊆ 1c
8	4	pw1ex 4303	. . . . . . . . . 10 ⊢ ℘1x ∈ V
9		1cex 4142	. . . . . . . . . 10 ⊢ 1c ∈ V
10	8, 9	nclec 6195	. . . . . . . . 9 ⊢ (℘1x ⊆ 1c → Nc ℘1x ≤c Nc 1c)
11	7, 10	ax-mp 5	. . . . . . . 8 ⊢ Nc ℘1x ≤c Nc 1c
12	6, 11	syl6eqbr 4676	. . . . . . 7 ⊢ (n = Nc x → Tc n ≤c Nc 1c)
13	12	exlimiv 1634	. . . . . 6 ⊢ (∃x n = Nc x → Tc n ≤c Nc 1c)
14	2, 13	sylbi 187	. . . . 5 ⊢ (n ∈ NC → Tc n ≤c Nc 1c)
15		breq1 4642	. . . . 5 ⊢ (M = Tc n → (M ≤c Nc 1c ↔ Tc n ≤c Nc 1c))
16	14, 15	syl5ibrcom 213	. . . 4 ⊢ (n ∈ NC → (M = Tc n → M ≤c Nc 1c))
17	16	rexlimiv 2732	. . 3 ⊢ (∃n ∈ NC M = Tc n → M ≤c Nc 1c)
18	9	lenc 6223	. . . 4 ⊢ (M ∈ NC → (M ≤c Nc 1c ↔ ∃x ∈ M x ⊆ 1c))
19		ncseqnc 6128	. . . . . . 7 ⊢ (M ∈ NC → (M = Nc x ↔ x ∈ M))
20	19	biimpar 471	. . . . . 6 ⊢ ((M ∈ NC ∧ x ∈ M) → M = Nc x)
21	4	sspw12 4336	. . . . . . . 8 ⊢ (x ⊆ 1c ↔ ∃y x = ℘1y)
22		vex 2862	. . . . . . . . . . . 12 ⊢ y ∈ V
23	22	ncelncsi 6121	. . . . . . . . . . 11 ⊢ Nc y ∈ NC
24	22	tcnc 6225	. . . . . . . . . . 11 ⊢ Tc Nc y = Nc ℘1y
25		tceq 6158	. . . . . . . . . . . . 13 ⊢ (n = Nc y → Tc n = Tc Nc y)
26	25	eqeq1d 2361	. . . . . . . . . . . 12 ⊢ (n = Nc y → ( Tc n = Nc ℘1y ↔ Tc Nc y = Nc ℘1y))
27	26	rspcev 2955	. . . . . . . . . . 11 ⊢ (( Nc y ∈ NC ∧ Tc Nc y = Nc ℘1y) → ∃n ∈ NC Tc n = Nc ℘1y)
28	23, 24, 27	mp2an 653	. . . . . . . . . 10 ⊢ ∃n ∈ NC Tc n = Nc ℘1y
29		nceq 6108	. . . . . . . . . . . . 13 ⊢ (x = ℘1y → Nc x = Nc ℘1y)
30	29	eqeq1d 2361	. . . . . . . . . . . 12 ⊢ (x = ℘1y → ( Nc x = Tc n ↔ Nc ℘1y = Tc n))
31		eqcom 2355	. . . . . . . . . . . 12 ⊢ ( Nc ℘1y = Tc n ↔ Tc n = Nc ℘1y)
32	30, 31	syl6bb 252	. . . . . . . . . . 11 ⊢ (x = ℘1y → ( Nc x = Tc n ↔ Tc n = Nc ℘1y))
33	32	rexbidv 2635	. . . . . . . . . 10 ⊢ (x = ℘1y → (∃n ∈ NC Nc x = Tc n ↔ ∃n ∈ NC Tc n = Nc ℘1y))
34	28, 33	mpbiri 224	. . . . . . . . 9 ⊢ (x = ℘1y → ∃n ∈ NC Nc x = Tc n)
35	34	exlimiv 1634	. . . . . . . 8 ⊢ (∃y x = ℘1y → ∃n ∈ NC Nc x = Tc n)
36	21, 35	sylbi 187	. . . . . . 7 ⊢ (x ⊆ 1c → ∃n ∈ NC Nc x = Tc n)
37		eqeq1 2359	. . . . . . . 8 ⊢ (M = Nc x → (M = Tc n ↔ Nc x = Tc n))
38	37	rexbidv 2635	. . . . . . 7 ⊢ (M = Nc x → (∃n ∈ NC M = Tc n ↔ ∃n ∈ NC Nc x = Tc n))
39	36, 38	syl5ibr 212	. . . . . 6 ⊢ (M = Nc x → (x ⊆ 1c → ∃n ∈ NC M = Tc n))
40	20, 39	syl 15	. . . . 5 ⊢ ((M ∈ NC ∧ x ∈ M) → (x ⊆ 1c → ∃n ∈ NC M = Tc n))
41	40	rexlimdva 2738	. . . 4 ⊢ (M ∈ NC → (∃x ∈ M x ⊆ 1c → ∃n ∈ NC M = Tc n))
42	18, 41	sylbid 206	. . 3 ⊢ (M ∈ NC → (M ≤c Nc 1c → ∃n ∈ NC M = Tc n))
43	17, 42	impbid2 195	. 2 ⊢ (M ∈ NC → (∃n ∈ NC M = Tc n ↔ M ≤c Nc 1c))
44	1, 43	bitrd 244	1 ⊢ (M ∈ NC → ((M ↑c 0c) ∈ NC ↔ M ≤c Nc 1c))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ⊆ wss 3257  1_cc1c 4134  ℘₁cpw1 4135  0_cc0c 4374   class class class wbr 4639  (class class class)co 5525   NC cncs 6088   ≤_c clec 6089   Nc cnc 6091   T_c ctc 6093   ↑_c cce 6096

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101  df-tc 6103  df-ce 6106

This theorem is referenced by:  nchoicelem8  6296  nchoicelem9  6297