Theorem nc0le1 6216

Description: Any cardinal is either zero or no greater than one. Theorem XI.2.24 of [Rosser] p. 377. (Contributed by SF, 12-Mar-2015.)

Assertion

Ref	Expression
nc0le1	⊢ (N ∈ NC → (N = 0c ∨ 1c ≤c N))

Proof of Theorem nc0le1

Dummy variables p a q x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elncs 6119	. 2 ⊢ (N ∈ NC ↔ ∃a N = Nc a)
2		nceq 6108	. . . . . . 7 ⊢ (a = ∅ → Nc a = Nc ∅)
3		df0c2 6137	. . . . . . 7 ⊢ 0c = Nc ∅
4	2, 3	syl6eqr 2403	. . . . . 6 ⊢ (a = ∅ → Nc a = 0c)
5	4	orcd 381	. . . . 5 ⊢ (a = ∅ → ( Nc a = 0c ∨ 1c ≤c Nc a))
6		vex 2862	. . . . . . . . . 10 ⊢ x ∈ V
7	6	snss 3838	. . . . . . . . 9 ⊢ (x ∈ a ↔ {x} ⊆ a)
8		vex 2862	. . . . . . . . . . 11 ⊢ a ∈ V
9	8	ncid 6123	. . . . . . . . . 10 ⊢ a ∈ Nc a
10	6	snel1c 4140	. . . . . . . . . 10 ⊢ {x} ∈ 1c
11		sseq2 3293	. . . . . . . . . . 11 ⊢ (p = a → (q ⊆ p ↔ q ⊆ a))
12		sseq1 3292	. . . . . . . . . . 11 ⊢ (q = {x} → (q ⊆ a ↔ {x} ⊆ a))
13	11, 12	rspc2ev 2963	. . . . . . . . . 10 ⊢ ((a ∈ Nc a ∧ {x} ∈ 1c ∧ {x} ⊆ a) → ∃p ∈ Nc a∃q ∈ 1c q ⊆ p)
14	9, 10, 13	mp3an12 1267	. . . . . . . . 9 ⊢ ({x} ⊆ a → ∃p ∈ Nc a∃q ∈ 1c q ⊆ p)
15	7, 14	sylbi 187	. . . . . . . 8 ⊢ (x ∈ a → ∃p ∈ Nc a∃q ∈ 1c q ⊆ p)
16	15	exlimiv 1634	. . . . . . 7 ⊢ (∃x x ∈ a → ∃p ∈ Nc a∃q ∈ 1c q ⊆ p)
17		n0 3559	. . . . . . 7 ⊢ (a ≠ ∅ ↔ ∃x x ∈ a)
18		1cex 4142	. . . . . . . . 9 ⊢ 1c ∈ V
19		ncex 6117	. . . . . . . . 9 ⊢ Nc a ∈ V
20	18, 19	brlec 6113	. . . . . . . 8 ⊢ (1c ≤c Nc a ↔ ∃q ∈ 1c ∃p ∈ Nc aq ⊆ p)
21		rexcom 2772	. . . . . . . 8 ⊢ (∃q ∈ 1c ∃p ∈ Nc aq ⊆ p ↔ ∃p ∈ Nc a∃q ∈ 1c q ⊆ p)
22	20, 21	bitri 240	. . . . . . 7 ⊢ (1c ≤c Nc a ↔ ∃p ∈ Nc a∃q ∈ 1c q ⊆ p)
23	16, 17, 22	3imtr4i 257	. . . . . 6 ⊢ (a ≠ ∅ → 1c ≤c Nc a)
24	23	olcd 382	. . . . 5 ⊢ (a ≠ ∅ → ( Nc a = 0c ∨ 1c ≤c Nc a))
25	5, 24	pm2.61ine 2592	. . . 4 ⊢ ( Nc a = 0c ∨ 1c ≤c Nc a)
26		eqeq1 2359	. . . . 5 ⊢ (N = Nc a → (N = 0c ↔ Nc a = 0c))
27		breq2 4643	. . . . 5 ⊢ (N = Nc a → (1c ≤c N ↔ 1c ≤c Nc a))
28	26, 27	orbi12d 690	. . . 4 ⊢ (N = Nc a → ((N = 0c ∨ 1c ≤c N) ↔ ( Nc a = 0c ∨ 1c ≤c Nc a)))
29	25, 28	mpbiri 224	. . 3 ⊢ (N = Nc a → (N = 0c ∨ 1c ≤c N))
30	29	exlimiv 1634	. 2 ⊢ (∃a N = Nc a → (N = 0c ∨ 1c ≤c N))
31	1, 30	sylbi 187	1 ⊢ (N ∈ NC → (N = 0c ∨ 1c ≤c N))

Syntax hints:   → wi 4   ∨ wo 357  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ⊆ wss 3257  ∅c0 3550  {csn 3737  1_cc1c 4134  0_cc0c 4374   class class class wbr 4639   NC cncs 6088   ≤_c clec 6089   Nc cnc 6091

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101

This theorem is referenced by:  nc0suc  6217  leconnnc  6218  ncslemuc  6255