Theorem ltcpw1pwg 6203

Description: The cardinality of a unit power class is strictly less than the cardinality of the power class. Theorem XI.2.17 of [Rosser] p. 376. (Contributed by SF, 10-Mar-2015.)

Assertion

Ref	Expression
ltcpw1pwg	|- (A e. V -> Nc ~P1 A <c Nc ~PA)

Proof of Theorem ltcpw1pwg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw1exg 4303	. . . . 5 |- (A e. V -> ~P1 A e. _V)
2		ncidg 6123	. . . . 5 |- (~P1 A e. _V -> ~P1 A e. Nc ~P1 A)
3	1, 2	syl 15	. . . 4 |- (A e. V -> ~P1 A e. Nc ~P1 A)
4		pwexg 4329	. . . . 5 |- (A e. V -> ~PA e. _V)
5		ncidg 6123	. . . . 5 |- (~PA e. _V -> ~PA e. Nc ~PA)
6	4, 5	syl 15	. . . 4 |- (A e. V -> ~PA e. Nc ~PA)
7		pw1sspw 4172	. . . . 5 |- ~P1 A C_ ~PA
8		sseq1 3293	. . . . . 6 |- (x = ~P1 A -> (x C_ y <-> ~P1 A C_ y))
9		sseq2 3294	. . . . . 6 |- (y = ~PA -> (~P1 A C_ y <-> ~P1 A C_ ~PA))
10	8, 9	rspc2ev 2964	. . . . 5 |- ((~P1 A e. Nc ~P1 A /\ ~PA e. Nc ~PA /\ ~P1 A C_ ~PA) -> E.x e. Nc ~P1 AE.y e. Nc ~PAx C_ y)
11	7, 10	mp3an3 1266	. . . 4 |- ((~P1 A e. Nc ~P1 A /\ ~PA e. Nc ~PA) -> E.x e. Nc ~P1 AE.y e. Nc ~PAx C_ y)
12	3, 6, 11	syl2anc 642	. . 3 |- (A e. V -> E.x e. Nc ~P1 AE.y e. Nc ~PAx C_ y)
13		ncex 6118	. . . 4 |- Nc ~P1 A e. _V
14		ncex 6118	. . . 4 |- Nc ~PA e. _V
15	13, 14	brlec 6114	. . 3 |- ( Nc ~P1 A <_c Nc ~PA <-> E.x e. Nc ~P1 AE.y e. Nc ~PAx C_ y)
16	12, 15	sylibr 203	. 2 |- (A e. V -> Nc ~P1 A <_c Nc ~PA)
17		ncpw1pwneg 6202	. 2 |- (A e. V -> Nc ~P1 A =/= Nc ~PA)
18		brltc 6115	. 2 |- ( Nc ~P1 A <c Nc ~PA <-> ( Nc ~P1 A <_c Nc ~PA /\ Nc ~P1 A =/= Nc ~PA))
19	16, 17, 18	sylanbrc 645	1 |- (A e. V -> Nc ~P1 A <c Nc ~PA)

Syntax hints:   -> wi 4   e. wcel 1710   =/= wne 2517  E.wrex 2616  _Vcvv 2860   C_ wss 3258  ~Pcpw 3723  ~P₁ cpw1 4136   class class class wbr 4640   <__c clec 6090   <_c cltc 6091   Nc cnc 6092

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-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-fullfun 5769  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-en 6030  df-lec 6100  df-ltc 6101  df-nc 6102

This theorem is referenced by:  ce2lt  6221  nchoicelem19  6308  canltpw  6335