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Theorem unxpdom 8208
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 8004 . . . 4 Rel ≺
21brrelex2i 5193 . . 3 (1𝑜𝐴𝐴 ∈ V)
31brrelex2i 5193 . . 3 (1𝑜𝐵𝐵 ∈ V)
42, 3anim12i 589 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 breq2 4689 . . . . 5 (𝑥 = 𝐴 → (1𝑜𝑥 ↔ 1𝑜𝐴))
65anbi1d 741 . . . 4 (𝑥 = 𝐴 → ((1𝑜𝑥 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝑦)))
7 uneq1 3793 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
8 xpeq1 5157 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
97, 8breq12d 4698 . . . 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 4689 . . . . 5 (𝑦 = 𝐵 → (1𝑜𝑦 ↔ 1𝑜𝐵))
1211anbi2d 740 . . . 4 (𝑦 = 𝐵 → ((1𝑜𝐴 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝐵)))
13 uneq2 3794 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
14 xpeq2 5163 . . . . 5 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
1513, 14breq12d 4698 . . . 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 2651 . . . 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 2651 . . . 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 8207 . . 3 ((1𝑜𝑥 ∧ 1𝑜𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦))
2010, 16, 19vtocl2g 3301 . 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 383   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  ifcif 4119  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  1𝑜c1o 7598  cdom 7995  csdm 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000
This theorem is referenced by:  unxpdom2  8209  sucxpdom  8210  cdaxpdom  9049
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