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Theorem xpdom3m 6483
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 6471 . . . . . . 7 ((𝐴𝑉𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
213adant2 960 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
32ensymd 6433 . . . . 5 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≈ (𝐴 × {𝑥}))
4 xpexg 4513 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
543adant3 961 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × 𝐵) ∈ V)
6 simp3 943 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝑥𝐵)
76snssd 3559 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → {𝑥} ⊆ 𝐵)
8 xpss2 4510 . . . . . . 7 ({𝑥} ⊆ 𝐵 → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
97, 8syl 14 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
10 ssdomg 6428 . . . . . 6 ((𝐴 × 𝐵) ∈ V → ((𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)))
115, 9, 10sylc 61 . . . . 5 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵))
12 endomtr 6440 . . . . 5 ((𝐴 ≈ (𝐴 × {𝑥}) ∧ (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) → 𝐴 ≼ (𝐴 × 𝐵))
133, 11, 12syl2anc 403 . . . 4 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
14133expia 1143 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
1514exlimdv 1744 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
16153impia 1138 1 ((𝐴𝑉𝐵𝑊 ∧ ∃𝑥 𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 922  wex 1424  wcel 1436  Vcvv 2614  wss 2986  {csn 3425   class class class wbr 3814   × cxp 4402  cen 6388  cdom 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-er 6225  df-en 6391  df-dom 6392
This theorem is referenced by: (None)
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