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Theorem cdaxpdom 8894
Description: Cartesian product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
cdaxpdom ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))

Proof of Theorem cdaxpdom
StepHypRef Expression
1 relsdom 7848 . . . . 5 Rel ≺
21brrelex2i 5083 . . . 4 (1𝑜𝐴𝐴 ∈ V)
31brrelex2i 5083 . . . 4 (1𝑜𝐵𝐵 ∈ V)
4 cdaval 8875 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
52, 3, 4syl2an 493 . . 3 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
6 0ex 4718 . . . . . . 7 ∅ ∈ V
7 xpsneng 7930 . . . . . . 7 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
82, 6, 7sylancl 693 . . . . . 6 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
9 sdomen2 7990 . . . . . 6 ((𝐴 × {∅}) ≈ 𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜𝐴))
108, 9syl 17 . . . . 5 (1𝑜𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜𝐴))
1110ibir 256 . . . 4 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
12 1on 7454 . . . . . . 7 1𝑜 ∈ On
13 xpsneng 7930 . . . . . . 7 ((𝐵 ∈ V ∧ 1𝑜 ∈ On) → (𝐵 × {1𝑜}) ≈ 𝐵)
143, 12, 13sylancl 693 . . . . . 6 (1𝑜𝐵 → (𝐵 × {1𝑜}) ≈ 𝐵)
15 sdomen2 7990 . . . . . 6 ((𝐵 × {1𝑜}) ≈ 𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜𝐵))
1614, 15syl 17 . . . . 5 (1𝑜𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜𝐵))
1716ibir 256 . . . 4 (1𝑜𝐵 → 1𝑜 ≺ (𝐵 × {1𝑜}))
18 unxpdom 8052 . . . 4 ((1𝑜 ≺ (𝐴 × {∅}) ∧ 1𝑜 ≺ (𝐵 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
1911, 17, 18syl2an 493 . . 3 ((1𝑜𝐴 ∧ 1𝑜𝐵) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
205, 19eqbrtrd 4605 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
21 xpen 8008 . . 3 (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
228, 14, 21syl2an 493 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
23 domentr 7901 . 2 (((𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ∧ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
2420, 22, 23syl2anc 691 1 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  1𝑜c1o 7440  cen 7838  cdom 7839  csdm 7840   +𝑐 ccda 8872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-cda 8873
This theorem is referenced by:  canthp1lem1  9353
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