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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 → ((Mc 0c) NCMc Nc 1c))

Proof of Theorem ce0lenc1
Dummy variables n x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ce0tb 6238 . 2 (M NC → ((Mc 0c) NCn NC M = Tc n))
2 elncs 6119 . . . . . 6 (n NCx n = Nc x)
3 tceq 6158 . . . . . . . . 9 (n = Nc xTc n = Tc Nc x)
4 vex 2862 . . . . . . . . . 10 x V
54tcnc 6225 . . . . . . . . 9 Tc Nc x = Nc 1x
63, 5syl6eq 2401 . . . . . . . 8 (n = Nc xTc n = Nc 1x)
7 pw1ss1c 4158 . . . . . . . . 9 1x 1c
84pw1ex 4303 . . . . . . . . . 10 1x V
9 1cex 4142 . . . . . . . . . 10 1c V
108, 9nclec 6195 . . . . . . . . 9 (1x 1cNc 1xc Nc 1c)
117, 10ax-mp 5 . . . . . . . 8 Nc 1xc Nc 1c
126, 11syl6eqbr 4676 . . . . . . 7 (n = Nc xTc nc Nc 1c)
1312exlimiv 1634 . . . . . 6 (x n = Nc xTc nc Nc 1c)
142, 13sylbi 187 . . . . 5 (n NCTc nc Nc 1c)
15 breq1 4642 . . . . 5 (M = Tc n → (Mc Nc 1cTc nc Nc 1c))
1614, 15syl5ibrcom 213 . . . 4 (n NC → (M = Tc nMc Nc 1c))
1716rexlimiv 2732 . . 3 (n NC M = Tc nMc Nc 1c)
189lenc 6223 . . . 4 (M NC → (Mc Nc 1cx M x 1c))
19 ncseqnc 6128 . . . . . . 7 (M NC → (M = Nc xx M))
2019biimpar 471 . . . . . 6 ((M NC x M) → M = Nc x)
214sspw12 4336 . . . . . . . 8 (x 1cy x = 1y)
22 vex 2862 . . . . . . . . . . . 12 y V
2322ncelncsi 6121 . . . . . . . . . . 11 Nc y NC
2422tcnc 6225 . . . . . . . . . . 11 Tc Nc y = Nc 1y
25 tceq 6158 . . . . . . . . . . . . 13 (n = Nc yTc n = Tc Nc y)
2625eqeq1d 2361 . . . . . . . . . . . 12 (n = Nc y → ( Tc n = Nc 1yTc Nc y = Nc 1y))
2726rspcev 2955 . . . . . . . . . . 11 (( Nc y NC Tc Nc y = Nc 1y) → n NC Tc n = Nc 1y)
2823, 24, 27mp2an 653 . . . . . . . . . 10 n NC Tc n = Nc 1y
29 nceq 6108 . . . . . . . . . . . . 13 (x = 1yNc x = Nc 1y)
3029eqeq1d 2361 . . . . . . . . . . . 12 (x = 1y → ( Nc x = Tc nNc 1y = Tc n))
31 eqcom 2355 . . . . . . . . . . . 12 ( Nc 1y = Tc nTc n = Nc 1y)
3230, 31syl6bb 252 . . . . . . . . . . 11 (x = 1y → ( Nc x = Tc nTc n = Nc 1y))
3332rexbidv 2635 . . . . . . . . . 10 (x = 1y → (n NC Nc x = Tc nn NC Tc n = Nc 1y))
3428, 33mpbiri 224 . . . . . . . . 9 (x = 1yn NC Nc x = Tc n)
3534exlimiv 1634 . . . . . . . 8 (y x = 1yn NC Nc x = Tc n)
3621, 35sylbi 187 . . . . . . 7 (x 1cn NC Nc x = Tc n)
37 eqeq1 2359 . . . . . . . 8 (M = Nc x → (M = Tc nNc x = Tc n))
3837rexbidv 2635 . . . . . . 7 (M = Nc x → (n NC M = Tc nn NC Nc x = Tc n))
3936, 38syl5ibr 212 . . . . . 6 (M = Nc x → (x 1cn NC M = Tc n))
4020, 39syl 15 . . . . 5 ((M NC x M) → (x 1cn NC M = Tc n))
4140rexlimdva 2738 . . . 4 (M NC → (x M x 1cn NC M = Tc n))
4218, 41sylbid 206 . . 3 (M NC → (Mc Nc 1cn NC M = Tc n))
4317, 42impbid2 195 . 2 (M NC → (n NC M = Tc nMc Nc 1c))
441, 43bitrd 244 1 (M NC → ((Mc 0c) NCMc Nc 1c))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ⊆ wss 3257  1cc1c 4134  ℘1cpw1 4135  0cc0c 4374   class class class wbr 4639  (class class class)co 5525   NC cncs 6088   ≤c clec 6089   Nc cnc 6091   Tc 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
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