Theorem cdaxpdom 9333

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 8235	. . . . 5 ⊢ Rel ≺
2	1	brrelex2i 5398	. . . 4 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
3	1	brrelex2i 5398	. . . 4 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V)
4		cdaval 9314	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1o})))
5	2, 3, 4	syl2an 589	. . 3 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1o})))
6		0ex 5016	. . . . . . 7 ⊢ ∅ ∈ V
7		xpsneng 8320	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
8	2, 6, 7	sylancl 580	. . . . . 6 ⊢ (1o ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
9		sdomen2 8380	. . . . . 6 ⊢ ((𝐴 × {∅}) ≈ 𝐴 → (1o ≺ (𝐴 × {∅}) ↔ 1o ≺ 𝐴))
10	8, 9	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → (1o ≺ (𝐴 × {∅}) ↔ 1o ≺ 𝐴))
11	10	ibir 260	. . . 4 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {∅}))
12		1on 7838	. . . . . . 7 ⊢ 1o ∈ On
13		xpsneng 8320	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 1o ∈ On) → (𝐵 × {1o}) ≈ 𝐵)
14	3, 12, 13	sylancl 580	. . . . . 6 ⊢ (1o ≺ 𝐵 → (𝐵 × {1o}) ≈ 𝐵)
15		sdomen2 8380	. . . . . 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 260	. . . 4 ⊢ (1o ≺ 𝐵 → 1o ≺ (𝐵 × {1o}))
18		unxpdom 8442	. . . 4 ⊢ ((1o ≺ (𝐴 × {∅}) ∧ 1o ≺ (𝐵 × {1o})) → ((𝐴 × {∅}) ∪ (𝐵 × {1o})) ≼ ((𝐴 × {∅}) × (𝐵 × {1o})))
19	11, 17, 18	syl2an 589	. . 3 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → ((𝐴 × {∅}) ∪ (𝐵 × {1o})) ≼ ((𝐴 × {∅}) × (𝐵 × {1o})))
20	5, 19	eqbrtrd 4897	. 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1o})))
21		xpen 8398	. . 3 ⊢ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1o})) ≈ (𝐴 × 𝐵))
22	8, 14, 21	syl2an 589	. 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1o})) ≈ (𝐴 × 𝐵))
23		domentr 8287	. 2 ⊢ (((𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1o})) ∧ ((𝐴 × {∅}) × (𝐵 × {1o})) ≈ (𝐴 × 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
24	20, 22, 23	syl2anc 579	1 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ∪ cun 3796  ∅c0 4146  {csn 4399   class class class wbr 4875   × cxp 5344  Oncon0 5967  (class class class)co 6910  1_oc1o 7824   ≈ cen 8225   ≼ cdom 8226   ≺ csdm 8227   +_𝑐 ccda 9311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-1o 7831  df-2o 7832  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-cda 9312

This theorem is referenced by:  canthp1lem1  9796