Theorem ce0t 6233

Description: If (M ↑_c 0_c) is a cardinal, then M is a T-raising of some cardinal. (Contributed by SF, 17-Mar-2015.)

Assertion

Ref	Expression
ce0t	|- ((M e. NC /\ (M ↑c 0c) e. NC ) -> E.n e. NC M = Tc n)

Distinct variable group:   n,M

Proof of Theorem ce0t

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ce0ncpw1 6186	. 2 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> E.x M = Nc ~P1 x)
2		vex 2863	. . . . . 6 |- x e. _V
3	2	ncelncsi 6122	. . . . 5 |- Nc x e. NC
4	2	tcnc 6226	. . . . . 6 |- Tc Nc x = Nc ~P1 x
5	4	eqcomi 2357	. . . . 5 |- Nc ~P1 x = Tc Nc x
6		tceq 6159	. . . . . . 7 |- (n = Nc x -> Tc n = Tc Nc x)
7	6	eqeq2d 2364	. . . . . 6 |- (n = Nc x -> ( Nc ~P1 x = Tc n <-> Nc ~P1 x = Tc Nc x))
8	7	rspcev 2956	. . . . 5 |- (( Nc x e. NC /\ Nc ~P1 x = Tc Nc x) -> E.n e. NC Nc ~P1 x = Tc n)
9	3, 5, 8	mp2an 653	. . . 4 |- E.n e. NC Nc ~P1 x = Tc n
10		eqeq1 2359	. . . . 5 |- (M = Nc ~P1 x -> (M = Tc n <-> Nc ~P1 x = Tc n))
11	10	rexbidv 2636	. . . 4 |- (M = Nc ~P1 x -> (E.n e. NC M = Tc n <-> E.n e. NC Nc ~P1 x = Tc n))
12	9, 11	mpbiri 224	. . 3 |- (M = Nc ~P1 x -> E.n e. NC M = Tc n)
13	12	exlimiv 1634	. 2 |- (E.x M = Nc ~P1 x -> E.n e. NC M = Tc n)
14	1, 13	syl 15	1 |- ((M e. NC /\ (M ↑c 0c) e. NC ) -> E.n e. NC M = Tc n)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  ~P₁ cpw1 4136  0_cc0c 4375  (class class class)co 5526   NC cncs 6089   Nc cnc 6092   T_c ctc 6094   ↑_c cce 6097

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-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  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-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-tc 6104  df-ce 6107

This theorem is referenced by:  ce2le  6234  ce0tb  6239