Theorem infeq5 8534

Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 8540.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
infeq5	|- ( E. x x C. U. x <-> om e. _V )

Proof of Theorem infeq5

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

Step	Hyp	Ref	Expression
1		df-pss 3590	. . . . 5 |- ( x C. U. x <-> ( x C_ U. x /\ x =/= U. x ) )
2		unieq 4444	. . . . . . . . . 10 |- ( x = (/) -> U. x = U. (/) )
3		uni0 4465	. . . . . . . . . 10 |- U. (/) = (/)
4	2, 3	syl6req 2673	. . . . . . . . 9 |- ( x = (/) -> (/) = U. x )
5		eqtr 2641	. . . . . . . . 9 |- ( ( x = (/) /\ (/) = U. x ) -> x = U. x )
6	4, 5	mpdan 702	. . . . . . . 8 |- ( x = (/) -> x = U. x )
7	6	necon3i 2826	. . . . . . 7 |- ( x =/= U. x -> x =/= (/) )
8	7	anim1i 592	. . . . . 6 |- ( ( x =/= U. x /\ x C_ U. x ) -> ( x =/= (/) /\ x C_ U. x ) )
9	8	ancoms 469	. . . . 5 |- ( ( x C_ U. x /\ x =/= U. x ) -> ( x =/= (/) /\ x C_ U. x ) )
10	1, 9	sylbi 207	. . . 4 |- ( x C. U. x -> ( x =/= (/) /\ x C_ U. x ) )
11	10	eximi 1762	. . 3 |- ( E. x x C. U. x -> E. x ( x =/= (/) /\ x C_ U. x ) )
12		eqid 2622	. . . . 5 |- ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
13		eqid 2622	. . . . 5 |- ( rec ( ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) , (/) ) |` om ) = ( rec ( ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) , (/) ) |` om )
14		vex 3203	. . . . 5 |- x e. _V
15	12, 13, 14, 14	inf3lem7 8531	. . . 4 |- ( ( x =/= (/) /\ x C_ U. x ) -> om e. _V )
16	15	exlimiv 1858	. . 3 |- ( E. x ( x =/= (/) /\ x C_ U. x ) -> om e. _V )
17	11, 16	syl 17	. 2 |- ( E. x x C. U. x -> om e. _V )
18		infeq5i 8533	. 2 |- ( om e. _V -> E. x x C. U. x )
19	17, 18	impbii 199	1 |- ( E. x x C. U. x <-> om e. _V )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   U.cuni 4436   |-> cmpt 4729   |` cres 5116   omcom 7065   reccrdg 7505

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497

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-reu 2919  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-iun 4522  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-pred 5680  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-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by: (None)