Theorem unxpwdom 9547

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 8945	. . . . 5 ⊢ Rel ≼
2	1	brrelex2i 5718	. . . 4 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
3		domeng 8955	. . . 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 4237	. . . . . 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 3923	. . . . . . 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 2812	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) = 𝑥)
12	6, 11	breqtrrd 5139	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)))
13		unxpwdom2 9546	. . . 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 4131	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
16	2	adantr 485	. . . . . . . 8 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐵 ∪ 𝐶) ∈ V)
17		ssexg 5290	. . . . . . . 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 4190	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
20		ssdomg 8993	. . . . . . 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 9532	. . . . . 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 9533	. . . . . 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 4132	. . . . . . 7 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶)
28		ssexg 5290	. . . . . . 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 4190	. . . . . 6 ⊢ (𝑥 ∩ 𝐶) ⊆ 𝐶
31		ssdomg 8993	. . . . . 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 9000	. . . . . 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 1965	1 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5109   × cxp 5659   ≈ cen 8936   ≼ cdom 8937   ≼^* cwdom 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1st 7982  df-2nd 7983  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-wdom 9523

This theorem is referenced by:  pwdjudom  10194