Theorem ce0nnul 6178

Description: A condition for cardinal exponentiation being nonempty. Theorem XI.2.42 of [Rosser] p. 382. (Contributed by SF, 6-Mar-2015.)

Assertion

Ref	Expression
ce0nnul	|- (M e. NC -> ((M ↑c 0c) =/= (/) <-> E.a~P1 a e. M))

Distinct variable group:   M,a

Proof of Theorem ce0nnul

Dummy variables b g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cnc 6139	. . . 4 |- 0c e. NC
2		elce 6176	. . . 4 |- ((M e. NC /\ 0c e. NC ) -> (g e. (M ↑c 0c) <-> E.aE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b))))
3	1, 2	mpan2 652	. . 3 |- (M e. NC -> (g e. (M ↑c 0c) <-> E.aE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b))))
4	3	exbidv 1626	. 2 |- (M e. NC -> (E.g g e. (M ↑c 0c) <-> E.gE.aE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b))))
5		n0 3560	. 2 |- ((M ↑c 0c) =/= (/) <-> E.g g e. (M ↑c 0c))
6		19.42vv 1907	. . . . 5 |- (E.gE.b(~P1 a e. M /\ (~P1 b e. 0c /\ g ~~ (a ^m b))) <-> (~P1 a e. M /\ E.gE.b(~P1 b e. 0c /\ g ~~ (a ^m b))))
7		3anass 938	. . . . . 6 |- ((~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b)) <-> (~P1 a e. M /\ (~P1 b e. 0c /\ g ~~ (a ^m b))))
8	7	2exbii 1583	. . . . 5 |- (E.gE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b)) <-> E.gE.b(~P1 a e. M /\ (~P1 b e. 0c /\ g ~~ (a ^m b))))
9		nulel0c 4423	. . . . . . 7 |- (/) e. 0c
10		ovex 5552	. . . . . . . 8 |- (a ^m (/)) e. _V
11	10	enrflx 6036	. . . . . . 7 |- (a ^m (/)) ~~ (a ^m (/))
12		0ex 4111	. . . . . . . 8 |- (/) e. _V
13		pw1eq 4144	. . . . . . . . . . . 12 |- (b = (/) -> ~P1 b = ~P1 (/))
14		pw10 4162	. . . . . . . . . . . 12 |- ~P1 (/) = (/)
15	13, 14	syl6eq 2401	. . . . . . . . . . 11 |- (b = (/) -> ~P1 b = (/))
16	15	eleq1d 2419	. . . . . . . . . 10 |- (b = (/) -> (~P1 b e. 0c <-> (/) e. 0c))
17	16	adantl 452	. . . . . . . . 9 |- ((g = (a ^m (/)) /\ b = (/)) -> (~P1 b e. 0c <-> (/) e. 0c))
18		id 19	. . . . . . . . . 10 |- (g = (a ^m (/)) -> g = (a ^m (/)))
19		oveq2 5532	. . . . . . . . . 10 |- (b = (/) -> (a ^m b) = (a ^m (/)))
20	18, 19	breqan12d 4655	. . . . . . . . 9 |- ((g = (a ^m (/)) /\ b = (/)) -> (g ~~ (a ^m b) <-> (a ^m (/)) ~~ (a ^m (/))))
21	17, 20	anbi12d 691	. . . . . . . 8 |- ((g = (a ^m (/)) /\ b = (/)) -> ((~P1 b e. 0c /\ g ~~ (a ^m b)) <-> ((/) e. 0c /\ (a ^m (/)) ~~ (a ^m (/)))))
22	10, 12, 21	spc2ev 2948	. . . . . . 7 |- (((/) e. 0c /\ (a ^m (/)) ~~ (a ^m (/))) -> E.gE.b(~P1 b e. 0c /\ g ~~ (a ^m b)))
23	9, 11, 22	mp2an 653	. . . . . 6 |- E.gE.b(~P1 b e. 0c /\ g ~~ (a ^m b))
24	23	biantru 491	. . . . 5 |- (~P1 a e. M <-> (~P1 a e. M /\ E.gE.b(~P1 b e. 0c /\ g ~~ (a ^m b))))
25	6, 8, 24	3bitr4ri 269	. . . 4 |- (~P1 a e. M <-> E.gE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b)))
26	25	exbii 1582	. . 3 |- (E.a~P1 a e. M <-> E.aE.gE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b)))
27		excom 1741	. . 3 |- (E.aE.gE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b)) <-> E.gE.aE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b)))
28	26, 27	bitri 240	. 2 |- (E.a~P1 a e. M <-> E.gE.aE.b(~P1 a e. M /\ ~P1 b e. 0c /\ g ~~ (a ^m b)))
29	4, 5, 28	3bitr4g 279	1 |- (M e. NC -> ((M ↑c 0c) =/= (/) <-> E.a~P1 a e. M))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  (/)c0 3551  ~P₁ cpw1 4136  0_cc0c 4375   class class class wbr 4640  (class class class)co 5526   ^m cmap 6000   ~~ cen 6029   NC cncs 6089   ↑_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-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-ce 6107

This theorem is referenced by:  ce0nnuli  6179  ce0addcnnul  6180  ce0nnulb  6183  ceclb  6184  ce0nulnc  6185  ce0ncpw1  6186  te0c  6238