MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpdom3 Structured version   Visualization version   GIF version

Theorem xpdom3 7920
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 3889 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
2 xpsneng 7907 . . . . . . . 8 ((𝐴𝑉𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
323adant2 1072 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≈ 𝐴)
43ensymd 7870 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≈ (𝐴 × {𝑥}))
5 xpexg 6835 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
653adant3 1073 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × 𝐵) ∈ V)
7 simp3 1055 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝑥𝐵)
87snssd 4280 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝑥𝐵) → {𝑥} ⊆ 𝐵)
9 xpss2 5141 . . . . . . . 8 ({𝑥} ⊆ 𝐵 → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
108, 9syl 17 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵))
11 ssdomg 7864 . . . . . . 7 ((𝐴 × 𝐵) ∈ V → ((𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)))
126, 10, 11sylc 62 . . . . . 6 ((𝐴𝑉𝐵𝑊𝑥𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵))
13 endomtr 7877 . . . . . 6 ((𝐴 ≈ (𝐴 × {𝑥}) ∧ (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) → 𝐴 ≼ (𝐴 × 𝐵))
144, 12, 13syl2anc 690 . . . . 5 ((𝐴𝑉𝐵𝑊𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
15143expia 1258 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
1615exlimdv 1847 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥𝐵𝐴 ≼ (𝐴 × 𝐵)))
171, 16syl5bi 230 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ≠ ∅ → 𝐴 ≼ (𝐴 × 𝐵)))
18173impia 1252 1 ((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wex 1694  wcel 1976  wne 2779  Vcvv 3172  wss 3539  c0 3873  {csn 4124   class class class wbr 4577   × cxp 5026  cen 7815  cdom 7816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-er 7606  df-en 7819  df-dom 7820
This theorem is referenced by:  mapdom2  7993  xpfir  8044  infxpabs  8894
  Copyright terms: Public domain W3C validator