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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 1cc N))

Proof of Theorem nc0le1
Dummy variables p a q x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . 2 (N NCa N = Nc a)
2 nceq 6108 . . . . . . 7 (a = Nc a = Nc )
3 df0c2 6137 . . . . . . 7 0c = Nc
42, 3syl6eqr 2403 . . . . . 6 (a = Nc a = 0c)
54orcd 381 . . . . 5 (a = → ( Nc a = 0c 1cc Nc a))
6 vex 2862 . . . . . . . . . 10 x V
76snss 3838 . . . . . . . . 9 (x a ↔ {x} a)
8 vex 2862 . . . . . . . . . . 11 a V
98ncid 6123 . . . . . . . . . 10 a Nc a
106snel1c 4140 . . . . . . . . . 10 {x} 1c
11 sseq2 3293 . . . . . . . . . . 11 (p = a → (q pq a))
12 sseq1 3292 . . . . . . . . . . 11 (q = {x} → (q a ↔ {x} a))
1311, 12rspc2ev 2963 . . . . . . . . . 10 ((a Nc a {x} 1c {x} a) → p Nc aq 1c q p)
149, 10, 13mp3an12 1267 . . . . . . . . 9 ({x} ap Nc aq 1c q p)
157, 14sylbi 187 . . . . . . . 8 (x ap Nc aq 1c q p)
1615exlimiv 1634 . . . . . . 7 (x x ap Nc aq 1c q p)
17 n0 3559 . . . . . . 7 (ax x a)
18 1cex 4142 . . . . . . . . 9 1c V
19 ncex 6117 . . . . . . . . 9 Nc a V
2018, 19brlec 6113 . . . . . . . 8 (1cc Nc aq 1c p Nc aq p)
21 rexcom 2772 . . . . . . . 8 (q 1c p Nc aq pp Nc aq 1c q p)
2220, 21bitri 240 . . . . . . 7 (1cc Nc ap Nc aq 1c q p)
2316, 17, 223imtr4i 257 . . . . . 6 (a → 1cc Nc a)
2423olcd 382 . . . . 5 (a → ( Nc a = 0c 1cc Nc a))
255, 24pm2.61ine 2592 . . . 4 ( Nc a = 0c 1cc Nc a)
26 eqeq1 2359 . . . . 5 (N = Nc a → (N = 0cNc a = 0c))
27 breq2 4643 . . . . 5 (N = Nc a → (1cc N ↔ 1cc Nc a))
2826, 27orbi12d 690 . . . 4 (N = Nc a → ((N = 0c 1cc N) ↔ ( Nc a = 0c 1cc Nc a)))
2925, 28mpbiri 224 . . 3 (N = Nc a → (N = 0c 1cc N))
3029exlimiv 1634 . 2 (a N = Nc a → (N = 0c 1cc N))
311, 30sylbi 187 1 (N NC → (N = 0c 1cc N))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ⊆ wss 3257  ∅c0 3550  {csn 3737  1cc1c 4134  0cc0c 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
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