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Theorem cdaxpdom 8996
 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 7947 . . . . 5 Rel ≺
21brrelex2i 5149 . . . 4 (1𝑜𝐴𝐴 ∈ V)
31brrelex2i 5149 . . . 4 (1𝑜𝐵𝐵 ∈ V)
4 cdaval 8977 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
52, 3, 4syl2an 494 . . 3 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
6 0ex 4781 . . . . . . 7 ∅ ∈ V
7 xpsneng 8030 . . . . . . 7 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
82, 6, 7sylancl 693 . . . . . 6 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
9 sdomen2 8090 . . . . . 6 ((𝐴 × {∅}) ≈ 𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜𝐴))
108, 9syl 17 . . . . 5 (1𝑜𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜𝐴))
1110ibir 257 . . . 4 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
12 1on 7552 . . . . . . 7 1𝑜 ∈ On
13 xpsneng 8030 . . . . . . 7 ((𝐵 ∈ V ∧ 1𝑜 ∈ On) → (𝐵 × {1𝑜}) ≈ 𝐵)
143, 12, 13sylancl 693 . . . . . 6 (1𝑜𝐵 → (𝐵 × {1𝑜}) ≈ 𝐵)
15 sdomen2 8090 . . . . . 6 ((𝐵 × {1𝑜}) ≈ 𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜𝐵))
1614, 15syl 17 . . . . 5 (1𝑜𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜𝐵))
1716ibir 257 . . . 4 (1𝑜𝐵 → 1𝑜 ≺ (𝐵 × {1𝑜}))
18 unxpdom 8152 . . . 4 ((1𝑜 ≺ (𝐴 × {∅}) ∧ 1𝑜 ≺ (𝐵 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
1911, 17, 18syl2an 494 . . 3 ((1𝑜𝐴 ∧ 1𝑜𝐵) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
205, 19eqbrtrd 4666 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
21 xpen 8108 . . 3 (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
228, 14, 21syl2an 494 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
23 domentr 8000 . 2 (((𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ∧ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
2420, 22, 23syl2anc 692 1 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  Vcvv 3195   ∪ cun 3565  ∅c0 3907  {csn 4168   class class class wbr 4644   × cxp 5102  Oncon0 5711  (class class class)co 6635  1𝑜c1o 7538   ≈ cen 7937   ≼ cdom 7938   ≺ csdm 7939   +𝑐 ccda 8974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-1o 7545  df-2o 7546  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-cda 8975 This theorem is referenced by:  canthp1lem1  9459
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