Theorem infpwfidom 7540

Description: The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption ( ~P A i^i Fin ) e. _V because this theorem also implies that A is a set if ~P A i^i Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
infpwfidom	|- ( ( ~P A i^i Fin ) e. _V -> A ~<_ ( ~P A i^i Fin ) )

Proof of Theorem infpwfidom

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

Step	Hyp	Ref	Expression
1		snelpwi 4346	. . 3 |- ( x e. A -> { x } e. ~P A )
2		snfig 7093	. . 3 |- ( x e. A -> { x } e. Fin )
3	1, 2	elind 3414	. 2 |- ( x e. A -> { x } e. ( ~P A i^i Fin ) )
4		sneqbg 3883	. . 3 |- ( x e. A -> ( { x } = { y } <-> x = y ) )
5	4	adantr 276	. 2 |- ( ( x e. A /\ y e. A ) -> ( { x } = { y } <-> x = y ) )
6	3, 5	dom2 7051	1 |- ( ( ~P A i^i Fin ) e. _V -> A ~<_ ( ~P A i^i Fin ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   ~Pcpw 3685   {csn 3705   class class class wbr 4125   ~<_ cdom 7011   Fincfn 7012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-en 7013  df-dom 7014  df-fin 7015

This theorem is referenced by: (None)