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Theorem xpdom3 9111
Description: A set is dominated by its Cartesian product with a nonempty set. Exercise 6 of [Suppes] p. 98. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xpdom3 ((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵))

Proof of Theorem xpdom3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 n0 4352 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
2 xpsneng 9097 . . . . . . . 8 ((𝐴𝑉𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
323adant2 1131 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
43ensymd 9046 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≈ (𝐴 × {𝑥}))
5 xpexg 7771 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
653adant3 1132 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × 𝐵) ∈ V)
7 simp3 1138 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝑥𝐵)
87snssd 4808 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝑥𝐵) → {𝑥} ⊆ 𝐵)
9 xpss2 5704 . . . . . . . 8 ({𝑥} ⊆ 𝐵 → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
108, 9syl 17 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
11 ssdomg 9041 . . . . . . 7 ((𝐴 × 𝐵) ∈ V → ((𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)))
126, 10, 11sylc 65 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵))
13 endomtr 9053 . . . . . 6 ((𝐴 ≈ (𝐴 × {𝑥}) ∧ (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) → 𝐴 ≼ (𝐴 × 𝐵))
144, 12, 13syl2anc 584 . . . . 5 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
15143expia 1121 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
1615exlimdv 1932 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
171, 16biimtrid 242 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ≠ ∅ → 𝐴 ≼ (𝐴 × 𝐵)))
18173impia 1117 1 ((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wex 1778  wcel 2107  wne 2939  Vcvv 3479  wss 3950  c0 4332  {csn 4625   class class class wbr 5142   × cxp 5682  cen 8983  cdom 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-er 8746  df-en 8987  df-dom 8988
This theorem is referenced by:  mapdom2  9189  xpfir  9301  infxpabs  10252
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