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Theorem unxpdom 8052
 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 7848 . . . 4 Rel ≺
21brrelex2i 5083 . . 3 (1𝑜𝐴𝐴 ∈ V)
31brrelex2i 5083 . . 3 (1𝑜𝐵𝐵 ∈ V)
42, 3anim12i 588 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 breq2 4587 . . . . 5 (𝑥 = 𝐴 → (1𝑜𝑥 ↔ 1𝑜𝐴))
65anbi1d 737 . . . 4 (𝑥 = 𝐴 → ((1𝑜𝑥 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝑦)))
7 uneq1 3722 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
8 xpeq1 5052 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
97, 8breq12d 4596 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴𝑦) ≼ (𝐴 × 𝑦)))
106, 9imbi12d 333 . . 3 (𝑥 = 𝐴 → (((1𝑜𝑥 ∧ 1𝑜𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1𝑜𝐴 ∧ 1𝑜𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦))))
11 breq2 4587 . . . . 5 (𝑦 = 𝐵 → (1𝑜𝑦 ↔ 1𝑜𝐵))
1211anbi2d 736 . . . 4 (𝑦 = 𝐵 → ((1𝑜𝐴 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝐵)))
13 uneq2 3723 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
14 xpeq2 5053 . . . . 5 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
1513, 14breq12d 4596 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴𝐵) ≼ (𝐴 × 𝐵)))
1612, 15imbi12d 333 . . 3 (𝑦 = 𝐵 → (((1𝑜𝐴 ∧ 1𝑜𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))))
17 eqid 2610 . . . 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 2610 . . . 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 8051 . . 3 ((1𝑜𝑥 ∧ 1𝑜𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦))
2010, 16, 19vtocl2g 3243 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵)))
214, 20mpcom 37 1 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  ifcif 4036  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  1𝑜c1o 7440   ≼ cdom 7839   ≺ csdm 7840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844 This theorem is referenced by:  unxpdom2  8053  sucxpdom  8054  cdaxpdom  8894
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