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Theorem infpwfidom 9969
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 5401 . . 3 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
2 snfi 8991 . . . 4 {𝑥} ∈ Fin
32a1i 11 . . 3 (𝑥𝐴 → {𝑥} ∈ Fin)
41, 3elind 4155 . 2 (𝑥𝐴 → {𝑥} ∈ (𝒫 𝐴 ∩ Fin))
5 sneqbg 4802 . . 3 (𝑥𝐴 → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
65adantr 482 . 2 ((𝑥𝐴𝑦𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
74, 6dom2 8938 1 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  Vcvv 3444  cin 3910  𝒫 cpw 4561  {csn 4587   class class class wbr 5106  cdom 8884  Fincfn 8886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-en 8887  df-dom 8888  df-fin 8890
This theorem is referenced by:  infpwfien  10003  ttukeylem1  10450  canthnum  10590
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