MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unxpdom Structured version   Visualization version   GIF version

Theorem unxpdom 8714
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 8505 . . . 4 Rel ≺
21brrelex2i 5608 . . 3 (1o𝐴𝐴 ∈ V)
31brrelex2i 5608 . . 3 (1o𝐵𝐵 ∈ V)
42, 3anim12i 612 . 2 ((1o𝐴 ∧ 1o𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 breq2 5067 . . . . 5 (𝑥 = 𝐴 → (1o𝑥 ↔ 1o𝐴))
65anbi1d 629 . . . 4 (𝑥 = 𝐴 → ((1o𝑥 ∧ 1o𝑦) ↔ (1o𝐴 ∧ 1o𝑦)))
7 uneq1 4136 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
8 xpeq1 5568 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
97, 8breq12d 5076 . . . 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 5067 . . . . 5 (𝑦 = 𝐵 → (1o𝑦 ↔ 1o𝐵))
1211anbi2d 628 . . . 4 (𝑦 = 𝐵 → ((1o𝐴 ∧ 1o𝑦) ↔ (1o𝐴 ∧ 1o𝐵)))
13 uneq2 4137 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
14 xpeq2 5575 . . . . 5 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
1513, 14breq12d 5076 . . . 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 2826 . . . 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 2826 . . . 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 8713 . . 3 ((1o𝑥 ∧ 1o𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦))
2010, 16, 19vtocl2g 3577 . 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 1530  wcel 2107  Vcvv 3500  cun 3938  ifcif 4470  cop 4570   class class class wbr 5063  cmpt 5143   × cxp 5552  1oc1o 8086  cdom 8496  csdm 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-1o 8093  df-2o 8094  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501
This theorem is referenced by:  unxpdom2  8715  sucxpdom  8716  djuxpdom  9600
  Copyright terms: Public domain W3C validator