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Theorem unxpdom 8713
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 8504 . . . 4 Rel ≺
21brrelex2i 5602 . . 3 (1o𝐴𝐴 ∈ V)
31brrelex2i 5602 . . 3 (1o𝐵𝐵 ∈ V)
42, 3anim12i 612 . 2 ((1o𝐴 ∧ 1o𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 breq2 5061 . . . . 5 (𝑥 = 𝐴 → (1o𝑥 ↔ 1o𝐴))
65anbi1d 629 . . . 4 (𝑥 = 𝐴 → ((1o𝑥 ∧ 1o𝑦) ↔ (1o𝐴 ∧ 1o𝑦)))
7 uneq1 4129 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
8 xpeq1 5562 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
97, 8breq12d 5070 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴𝑦) ≼ (𝐴 × 𝑦)))
106, 9imbi12d 346 . . 3 (𝑥 = 𝐴 → (((1o𝑥 ∧ 1o𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1o𝐴 ∧ 1o𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦))))
11 breq2 5061 . . . . 5 (𝑦 = 𝐵 → (1o𝑦 ↔ 1o𝐵))
1211anbi2d 628 . . . 4 (𝑦 = 𝐵 → ((1o𝐴 ∧ 1o𝑦) ↔ (1o𝐴 ∧ 1o𝐵)))
13 uneq2 4130 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
14 xpeq2 5569 . . . . 5 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
1513, 14breq12d 5070 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴𝐵) ≼ (𝐴 × 𝐵)))
1612, 15imbi12d 346 . . 3 (𝑦 = 𝐵 → (((1o𝐴 ∧ 1o𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1o𝐴 ∧ 1o𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))))
17 eqid 2818 . . . 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 2818 . . . 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 8712 . . 3 ((1o𝑥 ∧ 1o𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦))
2010, 16, 19vtocl2g 3569 . 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 396   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  ifcif 4463  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  1oc1o 8084  cdom 8495  csdm 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500
This theorem is referenced by:  unxpdom2  8714  sucxpdom  8715  djuxpdom  9599
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