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Theorem xpdom3m 6630
Description: A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
Assertion
Ref Expression
xpdom3m ((𝐴𝑉𝐵𝑊 ∧ ∃𝑥 𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑊

Proof of Theorem xpdom3m
StepHypRef Expression
1 xpsneng 6618 . . . . . . 7 ((𝐴𝑉𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
213adant2 965 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
32ensymd 6580 . . . . 5 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≈ (𝐴 × {𝑥}))
4 xpexg 4581 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
543adant3 966 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × 𝐵) ∈ V)
6 simp3 948 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝑥𝐵)
76snssd 3604 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → {𝑥} ⊆ 𝐵)
8 xpss2 4578 . . . . . . 7 ({𝑥} ⊆ 𝐵 → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
97, 8syl 14 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
10 ssdomg 6575 . . . . . 6 ((𝐴 × 𝐵) ∈ V → ((𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)))
115, 9, 10sylc 62 . . . . 5 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵))
12 endomtr 6587 . . . . 5 ((𝐴 ≈ (𝐴 × {𝑥}) ∧ (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) → 𝐴 ≼ (𝐴 × 𝐵))
133, 11, 12syl2anc 404 . . . 4 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
14133expia 1148 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
1514exlimdv 1754 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
16153impia 1143 1 ((𝐴𝑉𝐵𝑊 ∧ ∃𝑥 𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 927  wex 1433  wcel 1445  Vcvv 2633  wss 3013  {csn 3466   class class class wbr 3867   × cxp 4465  cen 6535  cdom 6536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-er 6332  df-en 6538  df-dom 6539
This theorem is referenced by: (None)
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