Theorem nc0le1 6217

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 6120	. 2 ⊢ (N ∈ NC ↔ ∃a N = Nc a)
2		nceq 6109	. . . . . . 7 ⊢ (a = ∅ → Nc a = Nc ∅)
3		df0c2 6138	. . . . . . 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 2863	. . . . . . . . . 10 ⊢ x ∈ V
7	6	snss 3839	. . . . . . . . 9 ⊢ (x ∈ a ↔ {x} ⊆ a)
8		vex 2863	. . . . . . . . . . 11 ⊢ a ∈ V
9	8	ncid 6124	. . . . . . . . . 10 ⊢ a ∈ Nc a
10	6	snel1c 4141	. . . . . . . . . 10 ⊢ {x} ∈ 1c
11		sseq2 3294	. . . . . . . . . . 11 ⊢ (p = a → (q ⊆ p ↔ q ⊆ a))
12		sseq1 3293	. . . . . . . . . . 11 ⊢ (q = {x} → (q ⊆ a ↔ {x} ⊆ a))
13	11, 12	rspc2ev 2964	. . . . . . . . . 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 3560	. . . . . . 7 ⊢ (a ≠ ∅ ↔ ∃x x ∈ a)
18		1cex 4143	. . . . . . . . 9 ⊢ 1c ∈ V
19		ncex 6118	. . . . . . . . 9 ⊢ Nc a ∈ V
20	18, 19	brlec 6114	. . . . . . . 8 ⊢ (1c ≤c Nc a ↔ ∃q ∈ 1c ∃p ∈ Nc aq ⊆ p)
21		rexcom 2773	. . . . . . . 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 2593	. . . 4 ⊢ ( Nc a = 0c ∨ 1c ≤c Nc a)
26		eqeq1 2359	. . . . 5 ⊢ (N = Nc a → (N = 0c ↔ Nc a = 0c))
27		breq2 4644	. . . . 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 2517  ∃wrex 2616   ⊆ wss 3258  ∅c0 3551  {csn 3738  1_cc1c 4135  0_cc0c 4375   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090   Nc cnc 6092

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-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102

This theorem is referenced by:  nc0suc  6218  leconnnc  6219  ncslemuc  6256