Theorem unxpwdom 9205

Description: If a Cartesian product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
unxpwdom	⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))

Proof of Theorem unxpwdom

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 8632	. . . . 5 ⊢ Rel ≼
2	1	brrelex2i 5606	. . . 4 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
3		domeng 8642	. . . 4 ⊢ ((𝐵 ∪ 𝐶) ∈ V → ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ↔ ∃𝑥((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))))
4	2, 3	syl 17	. . 3 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ↔ ∃𝑥((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))))
5	4	ibi 270	. 2 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → ∃𝑥((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶)))
6		simprl 771	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7		indi 4188	. . . . . 6 ⊢ (𝑥 ∩ (𝐵 ∪ 𝐶)) = ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶))
8		simprr 773	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝑥 ⊆ (𝐵 ∪ 𝐶))
9		df-ss 3883	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐵 ∪ 𝐶) ↔ (𝑥 ∩ (𝐵 ∪ 𝐶)) = 𝑥)
10	8, 9	sylib 221	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ (𝐵 ∪ 𝐶)) = 𝑥)
11	7, 10	eqtr3id 2792	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) = 𝑥)
12	6, 11	breqtrrd 5081	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)))
13		unxpwdom2 9204	. . . 4 ⊢ ((𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) → (𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)))
14	12, 13	syl 17	. . 3 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)))
15		ssun1 4086	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
16	2	adantr 484	. . . . . . . 8 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐵 ∪ 𝐶) ∈ V)
17		ssexg 5216	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐵 ∈ V)
18	15, 16, 17	sylancr 590	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐵 ∈ V)
19		inss2 4144	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
20		ssdomg 8674	. . . . . . 7 ⊢ (𝐵 ∈ V → ((𝑥 ∩ 𝐵) ⊆ 𝐵 → (𝑥 ∩ 𝐵) ≼ 𝐵))
21	18, 19, 20	mpisyl 21	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐵) ≼ 𝐵)
22		domwdom 9190	. . . . . 6 ⊢ ((𝑥 ∩ 𝐵) ≼ 𝐵 → (𝑥 ∩ 𝐵) ≼* 𝐵)
23	21, 22	syl 17	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐵) ≼* 𝐵)
24		wdomtr 9191	. . . . . 6 ⊢ ((𝐴 ≼* (𝑥 ∩ 𝐵) ∧ (𝑥 ∩ 𝐵) ≼* 𝐵) → 𝐴 ≼* 𝐵)
25	24	expcom 417	. . . . 5 ⊢ ((𝑥 ∩ 𝐵) ≼* 𝐵 → (𝐴 ≼* (𝑥 ∩ 𝐵) → 𝐴 ≼* 𝐵))
26	23, 25	syl 17	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* (𝑥 ∩ 𝐵) → 𝐴 ≼* 𝐵))
27		ssun2 4087	. . . . . . 7 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶)
28		ssexg 5216	. . . . . . 7 ⊢ ((𝐶 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐶 ∈ V)
29	27, 16, 28	sylancr 590	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐶 ∈ V)
30		inss2 4144	. . . . . 6 ⊢ (𝑥 ∩ 𝐶) ⊆ 𝐶
31		ssdomg 8674	. . . . . 6 ⊢ (𝐶 ∈ V → ((𝑥 ∩ 𝐶) ⊆ 𝐶 → (𝑥 ∩ 𝐶) ≼ 𝐶))
32	29, 30, 31	mpisyl 21	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐶) ≼ 𝐶)
33		domtr 8681	. . . . . 6 ⊢ ((𝐴 ≼ (𝑥 ∩ 𝐶) ∧ (𝑥 ∩ 𝐶) ≼ 𝐶) → 𝐴 ≼ 𝐶)
34	33	expcom 417	. . . . 5 ⊢ ((𝑥 ∩ 𝐶) ≼ 𝐶 → (𝐴 ≼ (𝑥 ∩ 𝐶) → 𝐴 ≼ 𝐶))
35	32, 34	syl 17	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼ (𝑥 ∩ 𝐶) → 𝐴 ≼ 𝐶))
36	26, 35	orim12d 965	. . 3 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → ((𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)))
37	14, 36	mpd 15	. 2 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))
38	5, 37	exlimddv 1943	1 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   = wceq 1543  ∃wex 1787   ∈ wcel 2110  Vcvv 3408   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866   class class class wbr 5053   × cxp 5549   ≈ cen 8623   ≼ cdom 8624   ≼^* cwdom 9180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1st 7761  df-2nd 7762  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-wdom 9181

This theorem is referenced by:  pwdjudom  9830