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

Theorem unxpdom 8029
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 7825 . . . 4 Rel ≺
21brrelex2i 5073 . . 3 (1𝑜𝐴𝐴 ∈ V)
31brrelex2i 5073 . . 3 (1𝑜𝐵𝐵 ∈ V)
42, 3anim12i 587 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 breq2 4581 . . . . 5 (𝑥 = 𝐴 → (1𝑜𝑥 ↔ 1𝑜𝐴))
65anbi1d 736 . . . 4 (𝑥 = 𝐴 → ((1𝑜𝑥 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝑦)))
7 uneq1 3721 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
8 xpeq1 5042 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
97, 8breq12d 4590 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴𝑦) ≼ (𝐴 × 𝑦)))
106, 9imbi12d 332 . . 3 (𝑥 = 𝐴 → (((1𝑜𝑥 ∧ 1𝑜𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1𝑜𝐴 ∧ 1𝑜𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦))))
11 breq2 4581 . . . . 5 (𝑦 = 𝐵 → (1𝑜𝑦 ↔ 1𝑜𝐵))
1211anbi2d 735 . . . 4 (𝑦 = 𝐵 → ((1𝑜𝐴 ∧ 1𝑜𝑦) ↔ (1𝑜𝐴 ∧ 1𝑜𝐵)))
13 uneq2 3722 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
14 xpeq2 5043 . . . . 5 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
1513, 14breq12d 4590 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴𝐵) ≼ (𝐴 × 𝐵)))
1612, 15imbi12d 332 . . 3 (𝑦 = 𝐵 → (((1𝑜𝐴 ∧ 1𝑜𝑦) → (𝐴𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))))
17 eqid 2609 . . . 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 2609 . . . 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 8028 . . 3 ((1𝑜𝑥 ∧ 1𝑜𝑦) → (𝑥𝑦) ≼ (𝑥 × 𝑦))
2010, 16, 19vtocl2g 3242 . 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 382   = wceq 1474  wcel 1976  Vcvv 3172  cun 3537  ifcif 4035  cop 4130   class class class wbr 4577  cmpt 4637   × cxp 5026  1𝑜c1o 7417  cdom 7816  csdm 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-1o 7424  df-2o 7425  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821
This theorem is referenced by:  unxpdom2  8030  sucxpdom  8031  cdaxpdom  8871
  Copyright terms: Public domain W3C validator