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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 NC c 0c NC c Nc 1c

Proof of Theorem ce0lenc1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ce0tb 6238 . 2 NC c 0c NC NC Tc
2 elncs 6119 . . . . . 6 NC Nc
3 tceq 6158 . . . . . . . . 9 Nc Tc Tc Nc
4 vex 2862 . . . . . . . . . 10
54tcnc 6225 . . . . . . . . 9 Tc Nc Nc 1
63, 5syl6eq 2401 . . . . . . . 8 Nc Tc Nc 1
7 pw1ss1c 4158 . . . . . . . . 9 1 1c
84pw1ex 4303 . . . . . . . . . 10 1
9 1cex 4142 . . . . . . . . . 10 1c
108, 9nclec 6195 . . . . . . . . 9 1 1c Nc 1 c Nc 1c
117, 10ax-mp 8 . . . . . . . 8 Nc 1 c Nc 1c
126, 11syl6eqbr 4676 . . . . . . 7 Nc Tc c Nc 1c
1312exlimiv 1634 . . . . . 6 Nc Tc c Nc 1c
142, 13sylbi 187 . . . . 5 NC Tc c Nc 1c
15 breq1 4642 . . . . 5 Tc c Nc 1c Tc c Nc 1c
1614, 15syl5ibrcom 213 . . . 4 NC Tc c Nc 1c
1716rexlimiv 2732 . . 3 NC Tc c Nc 1c
189lenc 6223 . . . 4 NC c Nc 1c 1c
19 ncseqnc 6128 . . . . . . 7 NC Nc
2019biimpar 471 . . . . . 6 NC Nc
214sspw12 4336 . . . . . . . 8 1c 1
22 vex 2862 . . . . . . . . . . . 12
2322ncelncsi 6121 . . . . . . . . . . 11 Nc NC
2422tcnc 6225 . . . . . . . . . . 11 Tc Nc Nc 1
25 tceq 6158 . . . . . . . . . . . . 13 Nc Tc Tc Nc
2625eqeq1d 2361 . . . . . . . . . . . 12 Nc Tc Nc 1 Tc Nc Nc 1
2726rspcev 2955 . . . . . . . . . . 11 Nc NC Tc Nc Nc 1 NC Tc Nc 1
2823, 24, 27mp2an 653 . . . . . . . . . 10 NC Tc Nc 1
29 nceq 6108 . . . . . . . . . . . . 13 1 Nc Nc 1
3029eqeq1d 2361 . . . . . . . . . . . 12 1 Nc Tc Nc 1 Tc
31 eqcom 2355 . . . . . . . . . . . 12 Nc 1 Tc Tc Nc 1
3230, 31syl6bb 252 . . . . . . . . . . 11 1 Nc Tc Tc Nc 1
3332rexbidv 2635 . . . . . . . . . 10 1 NC Nc Tc NC Tc Nc 1
3428, 33mpbiri 224 . . . . . . . . 9 1 NC Nc Tc
3534exlimiv 1634 . . . . . . . 8 1 NC Nc Tc
3621, 35sylbi 187 . . . . . . 7 1c NC Nc Tc
37 eqeq1 2359 . . . . . . . 8 Nc Tc Nc Tc
3837rexbidv 2635 . . . . . . 7 Nc NC Tc NC Nc Tc
3936, 38syl5ibr 212 . . . . . 6 Nc 1c NC Tc
4020, 39syl 15 . . . . 5 NC 1c NC Tc
4140rexlimdva 2738 . . . 4 NC 1c NC Tc
4218, 41sylbid 206 . . 3 NC c Nc 1c NC Tc
4317, 42impbid2 195 . 2 NC NC Tc c Nc 1c
441, 43bitrd 244 1 NC c 0c NC c 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  1 cpw1 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-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-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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