Theorem infeq5 8075

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 8081.) 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

Proof of Theorem infeq5

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

Step	Hyp	Ref	Expression
1		df-pss 3491	. . . . 5
2		unieq 4257	. . . . . . . . . 10
3		uni0 4276	. . . . . . . . . 10
4	2, 3	syl6req 2515	. . . . . . . . 9
5		eqtr 2483	. . . . . . . . 9
6	4, 5	mpdan 668	. . . . . . . 8
7	6	necon3i 2697	. . . . . . 7
8	7	anim1i 568	. . . . . 6
9	8	ancoms 453	. . . . 5
10	1, 9	sylbi 195	. . . 4
11	10	eximi 1656	. . 3
12		eqid 2457	. . . . 5
13		eqid 2457	. . . . 5
14		vex 3112	. . . . 5
15	12, 13, 14, 14	inf3lem7 8072	. . . 4
16	15	exlimiv 1722	. . 3
17	11, 16	syl 16	. 2
18		infeq5i 8074	. 2
19	17, 18	impbii 188	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652  {crab 2811  cvv 3109  i^icin 3474  C_wss 3475  C.wpss 3476  c0 3784  U.cuni 4249  e.cmpt 4510  |`cres 5006  com 6700  reccrdg 7094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-reg 8039

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6701  df-recs 7061  df-rdg 7095