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Theorem cdaxpdom 8866
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 7820 . . . . 5 Rel ≺
21brrelex2i 5068 . . . 4 (1𝑜𝐴𝐴 ∈ V)
31brrelex2i 5068 . . . 4 (1𝑜𝐵𝐵 ∈ V)
4 cdaval 8847 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
52, 3, 4syl2an 492 . . 3 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
6 0ex 4708 . . . . . . 7 ∅ ∈ V
7 xpsneng 7902 . . . . . . 7 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
82, 6, 7sylancl 692 . . . . . 6 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
9 sdomen2 7962 . . . . . 6 ((𝐴 × {∅}) ≈ 𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜𝐴))
108, 9syl 17 . . . . 5 (1𝑜𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜𝐴))
1110ibir 255 . . . 4 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
12 1on 7426 . . . . . . 7 1𝑜 ∈ On
13 xpsneng 7902 . . . . . . 7 ((𝐵 ∈ V ∧ 1𝑜 ∈ On) → (𝐵 × {1𝑜}) ≈ 𝐵)
143, 12, 13sylancl 692 . . . . . 6 (1𝑜𝐵 → (𝐵 × {1𝑜}) ≈ 𝐵)
15 sdomen2 7962 . . . . . 6 ((𝐵 × {1𝑜}) ≈ 𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜𝐵))
1614, 15syl 17 . . . . 5 (1𝑜𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜𝐵))
1716ibir 255 . . . 4 (1𝑜𝐵 → 1𝑜 ≺ (𝐵 × {1𝑜}))
18 unxpdom 8024 . . . 4 ((1𝑜 ≺ (𝐴 × {∅}) ∧ 1𝑜 ≺ (𝐵 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
1911, 17, 18syl2an 492 . . 3 ((1𝑜𝐴 ∧ 1𝑜𝐵) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
205, 19eqbrtrd 4594 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
21 xpen 7980 . . 3 (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
228, 14, 21syl2an 492 . 2 ((1𝑜𝐴 ∧ 1𝑜𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
23 domentr 7873 . 2 (((𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ∧ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
2420, 22, 23syl2anc 690 1 ((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  Vcvv 3167  cun 3532  c0 3868  {csn 4119   class class class wbr 4572   × cxp 5021  Oncon0 5621  (class class class)co 6522  1𝑜c1o 7412  cen 7810  cdom 7811  csdm 7812   +𝑐 ccda 8844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
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 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-1o 7419  df-2o 7420  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-cda 8845
This theorem is referenced by:  canthp1lem1  9325
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