Theorem infcntss 4567

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 (_ 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 4385	. 2 |- (om ~<_ A <-> E.x(om ~~ x /\ x (_ A))
3		visset 1816	. . . . . 6 |- x e. V
4	3	ensym 4418	. . . . 5 |- (om ~~ x -> x ~~ om)
5	4	anim2i 335	. . . 4 |- ((x (_ A /\ om ~~ x) -> (x (_ A /\ x ~~ om))
6	5	ancoms 438	. . 3 |- ((om ~~ x /\ x (_ A) -> (x (_ A /\ x ~~ om))
7	6	19.22i 1042	. 2 |- (E.x(om ~~ x /\ x (_ A) -> E.x(x (_ A /\ x ~~ om))
8	2, 7	sylbi 199	1 |- (om ~<_ A -> E.x(x (_ A /\ x ~~ om))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  E.wex 982  Vcvv 1814   (_ wss 2050   class class class wbr 2624  omcom 3137   ~~ cen 4370   ~<_ cdom 4371

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-er 4267  df-en 4374  df-dom 4375

Copyright terms: Public domain