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Theorem xpdom3 8611
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 4293 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
2 xpsneng 8598 . . . . . . . 8 ((𝐴𝑉𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
323adant2 1128 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
43ensymd 8556 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≈ (𝐴 × {𝑥}))
5 xpexg 7467 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
653adant3 1129 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × 𝐵) ∈ V)
7 simp3 1135 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝑥𝐵)
87snssd 4726 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝑥𝐵) → {𝑥} ⊆ 𝐵)
9 xpss2 5562 . . . . . . . 8 ({𝑥} ⊆ 𝐵 → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
108, 9syl 17 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
11 ssdomg 8551 . . . . . . 7 ((𝐴 × 𝐵) ∈ V → ((𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)))
126, 10, 11sylc 65 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵))
13 endomtr 8563 . . . . . 6 ((𝐴 ≈ (𝐴 × {𝑥}) ∧ (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) → 𝐴 ≼ (𝐴 × 𝐵))
144, 12, 13syl2anc 587 . . . . 5 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
15143expia 1118 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
1615exlimdv 1935 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
171, 16syl5bi 245 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ≠ ∅ → 𝐴 ≼ (𝐴 × 𝐵)))
18173impia 1114 1 ((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wex 1781  wcel 2115  wne 3014  Vcvv 3480  wss 3919  c0 4276  {csn 4550   class class class wbr 5052   × cxp 5540  cen 8502  cdom 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-er 8285  df-en 8506  df-dom 8507
This theorem is referenced by:  mapdom2  8685  xpfir  8737  infxpabs  9632
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