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Theorem cet 6234
 Description: The exponent of a T-raising to a T-raising is always a cardinal. (Contributed by SF, 13-Mar-2015.)
Assertion
Ref Expression
cet ((M NC N NC ) → ( Tc Mc Tc N) NC )

Proof of Theorem cet
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 tccl 6160 . . 3 (M NCTc M NC )
21adantr 451 . 2 ((M NC N NC ) → Tc M NC )
3 tccl 6160 . . 3 (N NCTc N NC )
43adantl 452 . 2 ((M NC N NC ) → Tc N NC )
5 elncs 6119 . . . 4 (M NCx M = Nc x)
6 tceq 6158 . . . . . . 7 (M = Nc xTc M = Tc Nc x)
76oveq1d 5537 . . . . . 6 (M = Nc x → ( Tc Mc 0c) = ( Tc Nc xc 0c))
8 vex 2862 . . . . . . . . . 10 x V
98ncelncsi 6121 . . . . . . . . 9 Nc x NC
10 tccl 6160 . . . . . . . . 9 ( Nc x NCTc Nc x NC )
119, 10ax-mp 5 . . . . . . . 8 Tc Nc x NC
128pw1ex 4303 . . . . . . . . . 10 1x V
1312ncid 6123 . . . . . . . . 9 1x Nc 1x
148tcnc 6225 . . . . . . . . 9 Tc Nc x = Nc 1x
1513, 14eleqtrri 2426 . . . . . . . 8 1x Tc Nc x
16 ce0nnuli 6178 . . . . . . . 8 (( Tc Nc x NC 1x Tc Nc x) → ( Tc Nc xc 0c) ≠ )
1711, 15, 16mp2an 653 . . . . . . 7 ( Tc Nc xc 0c) ≠
18 ce0nulnc 6184 . . . . . . . 8 ( Tc Nc x NC → (( Tc Nc xc 0c) ≠ ↔ ( Tc Nc xc 0c) NC ))
1911, 18ax-mp 5 . . . . . . 7 (( Tc Nc xc 0c) ≠ ↔ ( Tc Nc xc 0c) NC )
2017, 19mpbi 199 . . . . . 6 ( Tc Nc xc 0c) NC
217, 20syl6eqel 2441 . . . . 5 (M = Nc x → ( Tc Mc 0c) NC )
2221exlimiv 1634 . . . 4 (x M = Nc x → ( Tc Mc 0c) NC )
235, 22sylbi 187 . . 3 (M NC → ( Tc Mc 0c) NC )
2423adantr 451 . 2 ((M NC N NC ) → ( Tc Mc 0c) NC )
25 elncs 6119 . . . 4 (N NCx N = Nc x)
26 tceq 6158 . . . . . . 7 (N = Nc xTc N = Tc Nc x)
2726oveq1d 5537 . . . . . 6 (N = Nc x → ( Tc Nc 0c) = ( Tc Nc xc 0c))
2827, 20syl6eqel 2441 . . . . 5 (N = Nc x → ( Tc Nc 0c) NC )
2928exlimiv 1634 . . . 4 (x N = Nc x → ( Tc Nc 0c) NC )
3025, 29sylbi 187 . . 3 (N NC → ( Tc Nc 0c) NC )
3130adantl 452 . 2 ((M NC N NC ) → ( Tc Nc 0c) NC )
32 cecl 6186 . 2 ((( Tc M NC Tc N NC ) (( Tc Mc 0c) NC ( Tc Nc 0c) NC )) → ( Tc Mc Tc N) NC )
332, 4, 24, 31, 32syl22anc 1183 1 ((M NC N NC ) → ( Tc Mc Tc N) NC )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∅c0 3550  ℘1cpw1 4135  0cc0c 4374  (class class class)co 5525   NC cncs 6088   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-nc 6101  df-tc 6103  df-ce 6106 This theorem is referenced by:  ce2t  6235
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