Theorem infpss 9039

Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9135. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
infpss	|- ( om ~<_ A -> E. x ( x C. A /\ x ~~ A ) )

Distinct variable group:   x, A

Proof of Theorem infpss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infn0 8222	. . 3 |- ( om ~<_ A -> A =/= (/) )
2		n0 3931	. . 3 |- ( A =/= (/) <-> E. y y e. A )
3	1, 2	sylib 208	. 2 |- ( om ~<_ A -> E. y y e. A )
4		reldom 7961	. . . . . 6 |- Rel ~<_
5	4	brrelex2i 5159	. . . . 5 |- ( om ~<_ A -> A e. _V )
6		difexg 4808	. . . . 5 |- ( A e. _V -> ( A \ { y } ) e. _V )
7	5, 6	syl 17	. . . 4 |- ( om ~<_ A -> ( A \ { y } ) e. _V )
8	7	adantr 481	. . 3 |- ( ( om ~<_ A /\ y e. A ) -> ( A \ { y } ) e. _V )
9		simpr 477	. . . . 5 |- ( ( om ~<_ A /\ y e. A ) -> y e. A )
10		difsnpss 4338	. . . . 5 |- ( y e. A <-> ( A \ { y } ) C. A )
11	9, 10	sylib 208	. . . 4 |- ( ( om ~<_ A /\ y e. A ) -> ( A \ { y } ) C. A )
12		infdifsn 8554	. . . . 5 |- ( om ~<_ A -> ( A \ { y } ) ~~ A )
13	12	adantr 481	. . . 4 |- ( ( om ~<_ A /\ y e. A ) -> ( A \ { y } ) ~~ A )
14	11, 13	jca 554	. . 3 |- ( ( om ~<_ A /\ y e. A ) -> ( ( A \ { y } ) C. A /\ ( A \ { y } ) ~~ A ) )
15		psseq1 3694	. . . . 5 |- ( x = ( A \ { y } ) -> ( x C. A <-> ( A \ { y } ) C. A ) )
16		breq1 4656	. . . . 5 |- ( x = ( A \ { y } ) -> ( x ~~ A <-> ( A \ { y } ) ~~ A ) )
17	15, 16	anbi12d 747	. . . 4 |- ( x = ( A \ { y } ) -> ( ( x C. A /\ x ~~ A ) <-> ( ( A \ { y } ) C. A /\ ( A \ { y } ) ~~ A ) ) )
18	17	spcegv 3294	. . 3 |- ( ( A \ { y } ) e. _V -> ( ( ( A \ { y } ) C. A /\ ( A \ { y } ) ~~ A ) -> E. x ( x C. A /\ x ~~ A ) ) )
19	8, 14, 18	sylc 65	. 2 |- ( ( om ~<_ A /\ y e. A ) -> E. x ( x C. A /\ x ~~ A ) )
20	3, 19	exlimddv 1863	1 |- ( om ~<_ A -> E. x ( x C. A /\ x ~~ A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C. wpss 3575   (/)c0 3915   {csn 4177   class class class wbr 4653   omcom 7065   ~~ cen 7952   ~<_ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  isfin4-2  9136