Theorem infpwfidom 7175

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 4197	. . 3 |- ( x e. A -> { x } e. ~P A )
2		snfig 6792	. . 3 |- ( x e. A -> { x } e. Fin )
3	1, 2	elind 3312	. 2 |- ( x e. A -> { x } e. ( ~P A i^i Fin ) )
4		sneqbg 3750	. . 3 |- ( x e. A -> ( { x } = { y } <-> x = y ) )
5	4	adantr 274	. 2 |- ( ( x e. A /\ y e. A ) -> ( { x } = { y } <-> x = y ) )
6	3, 5	dom2 6753	1 |- ( ( ~P A i^i Fin ) e. _V -> A ~<_ ( ~P A i^i Fin ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   i^i cin 3120   ~Pcpw 3566   {csn 3583   class class class wbr 3989   ~<_ cdom 6717   Fincfn 6718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-en 6719  df-dom 6720  df-fin 6721

This theorem is referenced by: (None)