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Theorem unxpdom 9289
Description: Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
unxpdom ((1o𝐴 ∧ 1o𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))

Proof of Theorem unxpdom
Dummy variables 𝑥 𝑦 𝑢 𝑡 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsdom 8992 . . . 4 Rel ≺
21brrelex2i 5742 . . 3 (1o𝐴𝐴 ∈ V)
31brrelex2i 5742 . . 3 (1o𝐵𝐵 ∈ V)
42, 3anim12i 613 . 2 ((1o𝐴 ∧ 1o𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 breq2 5147 . . . . 5 (𝑥 = 𝐴 → (1o𝑥 ↔ 1o𝐴))
65anbi1d 631 . . . 4 (𝑥 = 𝐴 → ((1o𝑥 ∧ 1o𝑦) ↔ (1o𝐴 ∧ 1o𝑦)))
7 uneq1 4161 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
8 xpeq1 5699 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
97, 8breq12d 5156 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴𝑦) ≼ (𝐴 × 𝑦)))
106, 9imbi12d 344 . . 3 (𝑥 = 𝐴 → (((1o𝑥 ∧ 1o𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1o𝐴 ∧ 1o𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦))))
11 breq2 5147 . . . . 5 (𝑦 = 𝐵 → (1o𝑦 ↔ 1o𝐵))
1211anbi2d 630 . . . 4 (𝑦 = 𝐵 → ((1o𝐴 ∧ 1o𝑦) ↔ (1o𝐴 ∧ 1o𝐵)))
13 uneq2 4162 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
14 xpeq2 5706 . . . . 5 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
1513, 14breq12d 5156 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴𝐵) ≼ (𝐴 × 𝐵)))
1612, 15imbi12d 344 . . 3 (𝑦 = 𝐵 → (((1o𝐴 ∧ 1o𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1o𝐴 ∧ 1o𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))))
17 eqid 2737 . . . 4 (𝑧 ∈ (𝑥𝑦) ↦ if(𝑧𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩)) = (𝑧 ∈ (𝑥𝑦) ↦ if(𝑧𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩))
18 eqid 2737 . . . 4 if(𝑧𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩) = if(𝑧𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩)
1917, 18unxpdomlem3 9288 . . 3 ((1o𝑥 ∧ 1o𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦))
2010, 16, 19vtocl2g 3574 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1o𝐴 ∧ 1o𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵)))
214, 20mpcom 38 1 ((1o𝐴 ∧ 1o𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  ifcif 4525  cop 4632   class class class wbr 5143  cmpt 5225   × cxp 5683  1oc1o 8499  cdom 8983  csdm 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-2o 8507  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by:  unxpdom2  9290  sucxpdom  9291  djuxpdom  10226
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