Theorem unxpwdom 9551

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 8949	. . . . 5 ⊢ Rel ≼
2	1	brrelex2i 5719	. . . 4 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
3		domeng 8959	. . . 4 ⊢ ((𝐵 ∪ 𝐶) ∈ V → ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ↔ ∃𝑥((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))))
4	2, 3	syl 18	. . 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 782	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7		indi 4245	. . . . . 6 ⊢ (𝑥 ∩ (𝐵 ∪ 𝐶)) = ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶))
8		simprr 784	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝑥 ⊆ (𝐵 ∪ 𝐶))
9		dfss2 3931	. . . . . . 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 2818	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) = 𝑥)
12	6, 11	breqtrrd 5143	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)))
13		unxpwdom2 9550	. . . 4 ⊢ ((𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) → (𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)))
14	12, 13	syl 18	. . 3 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)))
15		ssun1 4139	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
16	2	adantr 485	. . . . . . . 8 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐵 ∪ 𝐶) ∈ V)
17		ssexg 5294	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐵 ∈ V)
18	15, 16, 17	sylancr 598	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐵 ∈ V)
19		inss2 4198	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
20		ssdomg 8997	. . . . . . 7 ⊢ (𝐵 ∈ V → ((𝑥 ∩ 𝐵) ⊆ 𝐵 → (𝑥 ∩ 𝐵) ≼ 𝐵))
21	18, 19, 20	mpisyl 22	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐵) ≼ 𝐵)
22		domwdom 9536	. . . . . 6 ⊢ ((𝑥 ∩ 𝐵) ≼ 𝐵 → (𝑥 ∩ 𝐵) ≼* 𝐵)
23	21, 22	syl 18	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐵) ≼* 𝐵)
24		wdomtr 9537	. . . . . 6 ⊢ ((𝐴 ≼* (𝑥 ∩ 𝐵) ∧ (𝑥 ∩ 𝐵) ≼* 𝐵) → 𝐴 ≼* 𝐵)
25	24	expcom 418	. . . . 5 ⊢ ((𝑥 ∩ 𝐵) ≼* 𝐵 → (𝐴 ≼* (𝑥 ∩ 𝐵) → 𝐴 ≼* 𝐵))
26	23, 25	syl 18	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* (𝑥 ∩ 𝐵) → 𝐴 ≼* 𝐵))
27		ssun2 4140	. . . . . . 7 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶)
28		ssexg 5294	. . . . . . 7 ⊢ ((𝐶 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐶 ∈ V)
29	27, 16, 28	sylancr 598	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐶 ∈ V)
30		inss2 4198	. . . . . 6 ⊢ (𝑥 ∩ 𝐶) ⊆ 𝐶
31		ssdomg 8997	. . . . . 6 ⊢ (𝐶 ∈ V → ((𝑥 ∩ 𝐶) ⊆ 𝐶 → (𝑥 ∩ 𝐶) ≼ 𝐶))
32	29, 30, 31	mpisyl 22	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐶) ≼ 𝐶)
33		domtr 9004	. . . . . 6 ⊢ ((𝐴 ≼ (𝑥 ∩ 𝐶) ∧ (𝑥 ∩ 𝐶) ≼ 𝐶) → 𝐴 ≼ 𝐶)
34	33	expcom 418	. . . . 5 ⊢ ((𝑥 ∩ 𝐶) ≼ 𝐶 → (𝐴 ≼ (𝑥 ∩ 𝐶) → 𝐴 ≼ 𝐶))
35	32, 34	syl 18	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼ (𝑥 ∩ 𝐶) → 𝐴 ≼ 𝐶))
36	26, 35	orim12d 979	. . 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 16	. 2 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))
38	5, 37	exlimddv 1962	1 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913   class class class wbr 5113   × cxp 5660   ≈ cen 8940   ≼ cdom 8941   ≼^* cwdom 9526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7986  df-2nd 7987  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-wdom 9527

This theorem is referenced by:  pwdjudom  10198