Theorem infcntss 5646

Description: Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.)

Hypothesis

Ref	Expression
infcntss.1	|- A e. _V

Assertion

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

Distinct variable group:   x,A

Proof of Theorem infcntss

Step	Hyp	Ref	Expression
1		infcntss.1	. . 3 |- A e. _V
2	1	domen 5438	. 2 |- (om ~<_ A <-> E.x(om ~~ x /\ x C_ A))
3		visset 2295	. . . . . 6 |- x e. _V
4	3	ensym 5471	. . . . 5 |- (om ~~ x -> x ~~ om)
5	4	anim2i 362	. . . 4 |- ((x C_ A /\ om ~~ x) -> (x C_ A /\ x ~~ om))
6	5	ancoms 484	. . 3 |- ((om ~~ x /\ x C_ A) -> (x C_ A /\ x ~~ om))
7	6	eximi 1387	. 2 |- (E.x(om ~~ x /\ x C_ A) -> E.x(x C_ A /\ x ~~ om))
8	2, 7	sylbi 216	1 |- (om ~<_ A -> E.x(x C_ A /\ x ~~ om))

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  E.wex 1326  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  omcom 3949   ~~ cen 5423   ~<_ cdom 5424

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-er 5318  df-en 5427  df-dom 5428

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