Theorem infpwfidom 7399

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 4301	. . 3 |- ( x e. A -> { x } e. ~P A )
2		snfig 6984	. . 3 |- ( x e. A -> { x } e. Fin )
3	1, 2	elind 3390	. 2 |- ( x e. A -> { x } e. ( ~P A i^i Fin ) )
4		sneqbg 3844	. . 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 6943	1 |- ( ( ~P A i^i Fin ) e. _V -> A ~<_ ( ~P A i^i Fin ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   i^i cin 3197   ~Pcpw 3650   {csn 3667   class class class wbr 4086   ~<_ cdom 6903   Fincfn 6904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-en 6905  df-dom 6906  df-fin 6907

This theorem is referenced by: (None)