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Theorem unxpdom 8374
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 8167 . . . 4 Rel ≺
21brrelex2i 5329 . . 3 (1𝑜𝐴𝐴 ∈ V)
31brrelex2i 5329 . . 3 (1𝑜𝐵𝐵 ∈ V)
42, 3anim12i 606 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 breq2 4813 . . . . 5 (𝑥 = 𝐴 → (1𝑜𝑥 ↔ 1𝑜𝐴))
65anbi1d 623 . . . 4 (𝑥 = 𝐴 → ((1𝑜𝑥 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝑦)))
7 uneq1 3922 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
8 xpeq1 5291 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
97, 8breq12d 4822 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴𝑦) ≼ (𝐴 × 𝑦)))
106, 9imbi12d 335 . . 3 (𝑥 = 𝐴 → (((1𝑜𝑥 ∧ 1𝑜𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1𝑜𝐴 ∧ 1𝑜𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦))))
11 breq2 4813 . . . . 5 (𝑦 = 𝐵 → (1𝑜𝑦 ↔ 1𝑜𝐵))
1211anbi2d 622 . . . 4 (𝑦 = 𝐵 → ((1𝑜𝐴 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝐵)))
13 uneq2 3923 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
14 xpeq2 5298 . . . . 5 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
1513, 14breq12d 4822 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴𝐵) ≼ (𝐴 × 𝐵)))
1612, 15imbi12d 335 . . 3 (𝑦 = 𝐵 → (((1𝑜𝐴 ∧ 1𝑜𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))))
17 eqid 2765 . . . 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 2765 . . . 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 8373 . . 3 ((1𝑜𝑥 ∧ 1𝑜𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦))
2010, 16, 19vtocl2g 3422 . 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 384   = wceq 1652  wcel 2155  Vcvv 3350  cun 3730  ifcif 4243  cop 4340   class class class wbr 4809  cmpt 4888   × cxp 5275  1𝑜c1o 7757  cdom 8158  csdm 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-2o 7765  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163
This theorem is referenced by:  unxpdom2  8375  sucxpdom  8376  cdaxpdom  9264
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