Theorem pw1eltc 6162

Description: The unit power class of an element of a cardinal is in the cardinal's T raising. (Contributed by SF, 2-Mar-2015.)

Assertion

Ref	Expression
pw1eltc	|- ((A e. NC /\ B e. A) -> ~P1 B e. Tc A)

Proof of Theorem pw1eltc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw1exg 4302	. . . 4 |- (B e. A -> ~P1 B e. _V)
2		ncidg 6122	. . . 4 |- (~P1 B e. _V -> ~P1 B e. Nc ~P1 B)
3	1, 2	syl 15	. . 3 |- (B e. A -> ~P1 B e. Nc ~P1 B)
4	3	adantl 452	. 2 |- ((A e. NC /\ B e. A) -> ~P1 B e. Nc ~P1 B)
5		eqid 2353	. . . . 5 |- Nc ~P1 B = Nc ~P1 B
6		pw1eq 4143	. . . . . . . 8 |- (y = B -> ~P1 y = ~P1 B)
7	6	nceqd 6110	. . . . . . 7 |- (y = B -> Nc ~P1 y = Nc ~P1 B)
8	7	eqeq2d 2364	. . . . . 6 |- (y = B -> ( Nc ~P1 B = Nc ~P1 y <-> Nc ~P1 B = Nc ~P1 B))
9	8	rspcev 2955	. . . . 5 |- ((B e. A /\ Nc ~P1 B = Nc ~P1 B) -> E.y e. A Nc ~P1 B = Nc ~P1 y)
10	5, 9	mpan2 652	. . . 4 |- (B e. A -> E.y e. A Nc ~P1 B = Nc ~P1 y)
11	10	adantl 452	. . 3 |- ((A e. NC /\ B e. A) -> E.y e. A Nc ~P1 B = Nc ~P1 y)
12		eqtc 6161	. . . 4 |- (A e. NC -> ( Tc A = Nc ~P1 B <-> E.y e. A Nc ~P1 B = Nc ~P1 y))
13	12	adantr 451	. . 3 |- ((A e. NC /\ B e. A) -> ( Tc A = Nc ~P1 B <-> E.y e. A Nc ~P1 B = Nc ~P1 y))
14	11, 13	mpbird 223	. 2 |- ((A e. NC /\ B e. A) -> Tc A = Nc ~P1 B)
15	4, 14	eleqtrrd 2430	1 |- ((A e. NC /\ B e. A) -> ~P1 B e. Tc A)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859  ~P₁ cpw1 4135   NC cncs 6088   Nc cnc 6091   T_c ctc 6093

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-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-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-tc 6103

This theorem is referenced by:  tc0c  6163  tcdi  6164  tc1c  6165  nntccl  6170  tcncg  6224