Theorem cdaxpdom 9011

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	⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))

Proof of Theorem cdaxpdom

Step	Hyp	Ref	Expression
1		relsdom 7962	. . . . 5 ⊢ Rel ≺
2	1	brrelex2i 5159	. . . 4 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V)
3	1	brrelex2i 5159	. . . 4 ⊢ (1𝑜 ≺ 𝐵 → 𝐵 ∈ V)
4		cdaval 8992	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
5	2, 3, 4	syl2an 494	. . 3 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
6		0ex 4790	. . . . . . 7 ⊢ ∅ ∈ V
7		xpsneng 8045	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
8	2, 6, 7	sylancl 694	. . . . . 6 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
9		sdomen2 8105	. . . . . 6 ⊢ ((𝐴 × {∅}) ≈ 𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜 ≺ 𝐴))
10	8, 9	syl 17	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜 ≺ 𝐴))
11	10	ibir 257	. . . 4 ⊢ (1𝑜 ≺ 𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
12		1on 7567	. . . . . . 7 ⊢ 1𝑜 ∈ On
13		xpsneng 8045	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 1𝑜 ∈ On) → (𝐵 × {1𝑜}) ≈ 𝐵)
14	3, 12, 13	sylancl 694	. . . . . 6 ⊢ (1𝑜 ≺ 𝐵 → (𝐵 × {1𝑜}) ≈ 𝐵)
15		sdomen2 8105	. . . . . 6 ⊢ ((𝐵 × {1𝑜}) ≈ 𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜 ≺ 𝐵))
16	14, 15	syl 17	. . . . 5 ⊢ (1𝑜 ≺ 𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜 ≺ 𝐵))
17	16	ibir 257	. . . 4 ⊢ (1𝑜 ≺ 𝐵 → 1𝑜 ≺ (𝐵 × {1𝑜}))
18		unxpdom 8167	. . . 4 ⊢ ((1𝑜 ≺ (𝐴 × {∅}) ∧ 1𝑜 ≺ (𝐵 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
19	11, 17, 18	syl2an 494	. . 3 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
20	5, 19	eqbrtrd 4675	. 2 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})))
21		xpen 8123	. . 3 ⊢ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
22	8, 14, 21	syl2an 494	. 2 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵))
23		domentr 8015	. 2 ⊢ (((𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ∧ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
24	20, 22, 23	syl2anc 693	1 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  Oncon0 5723  (class class class)co 6650  1_𝑜c1o 7553   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-cda 8990

This theorem is referenced by:  canthp1lem1  9474