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Theorem infpwfidom 7199
Description: The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
infpwfidom ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))

Proof of Theorem infpwfidom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snelpwi 4214 . . 3 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
2 snfig 6816 . . 3 (𝑥𝐴 → {𝑥} ∈ Fin)
31, 2elind 3322 . 2 (𝑥𝐴 → {𝑥} ∈ (𝒫 𝐴 ∩ Fin))
4 sneqbg 3765 . . 3 (𝑥𝐴 → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
54adantr 276 . 2 ((𝑥𝐴𝑦𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
63, 5dom2 6777 1 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  Vcvv 2739  cin 3130  𝒫 cpw 3577  {csn 3594   class class class wbr 4005  cdom 6741  Fincfn 6742
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1o 6419  df-en 6743  df-dom 6744  df-fin 6745
This theorem is referenced by: (None)
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