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Theorem xpdom3m 6244
 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 ((A 𝑉 B 𝑊 x x B) → A ≼ (A × B))
Distinct variable groups:   x,A   x,B   x,𝑉   x,𝑊

Proof of Theorem xpdom3m
StepHypRef Expression
1 xpsneng 6232 . . . . . . 7 ((A 𝑉 x B) → (A × {x}) ≈ A)
213adant2 922 . . . . . 6 ((A 𝑉 B 𝑊 x B) → (A × {x}) ≈ A)
32ensymd 6199 . . . . 5 ((A 𝑉 B 𝑊 x B) → A ≈ (A × {x}))
4 xpexg 4395 . . . . . . 7 ((A 𝑉 B 𝑊) → (A × B) V)
543adant3 923 . . . . . 6 ((A 𝑉 B 𝑊 x B) → (A × B) V)
6 simp3 905 . . . . . . . 8 ((A 𝑉 B 𝑊 x B) → x B)
76snssd 3500 . . . . . . 7 ((A 𝑉 B 𝑊 x B) → {x} ⊆ B)
8 xpss2 4392 . . . . . . 7 ({x} ⊆ B → (A × {x}) ⊆ (A × B))
97, 8syl 14 . . . . . 6 ((A 𝑉 B 𝑊 x B) → (A × {x}) ⊆ (A × B))
10 ssdomg 6194 . . . . . 6 ((A × B) V → ((A × {x}) ⊆ (A × B) → (A × {x}) ≼ (A × B)))
115, 9, 10sylc 56 . . . . 5 ((A 𝑉 B 𝑊 x B) → (A × {x}) ≼ (A × B))
12 endomtr 6206 . . . . 5 ((A ≈ (A × {x}) (A × {x}) ≼ (A × B)) → A ≼ (A × B))
133, 11, 12syl2anc 391 . . . 4 ((A 𝑉 B 𝑊 x B) → A ≼ (A × B))
14133expia 1105 . . 3 ((A 𝑉 B 𝑊) → (x BA ≼ (A × B)))
1514exlimdv 1697 . 2 ((A 𝑉 B 𝑊) → (x x BA ≼ (A × B)))
16153impia 1100 1 ((A 𝑉 B 𝑊 x x B) → A ≼ (A × B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  {csn 3367   class class class wbr 3755   × cxp 4286   ≈ cen 6155   ≼ cdom 6156 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-er 6042  df-en 6158  df-dom 6159 This theorem is referenced by: (None)
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