Theorem unxpdom2 3651

Description: Corollary of unxpdom 3650.

Hypotheses

Ref	Expression
unxpdom2.1	⊢ A ∈ V
unxpdom2.2	⊢ B ∈ V

Assertion

Ref	Expression
unxpdom2	⊢ ((1o ≺ A ∧ B ≼ A) → (A ∪ B) ≼ (A × A))

Proof of Theorem unxpdom2

Step	Hyp	Ref	Expression
1		domtr 3320	. . 3 ⊢ (((A ∪ B) ≼ ((A × {∅}) ∪ (A × {1o})) ∧ ((A × {∅}) ∪ (A × {1o})) ≼ (A × A)) → (A ∪ B) ≼ (A × A))
2		unxpdom2.1	. . . . . 6 ⊢ A ∈ V
3		1onn 3193	. . . . . . . 8 ⊢ 1o ∈ ω
4	3	elisseti 1355	. . . . . . 7 ⊢ 1o ∈ V
5	2, 4	xpsnen 3339	. . . . . 6 ⊢ (A × {1o}) ≈ A
6	2, 5	ensymi 3318	. . . . 5 ⊢ A ≈ (A × {1o})
7		domentr 3326	. . . . 5 ⊢ ((B ≼ A ∧ A ≈ (A × {1o})) → B ≼ (A × {1o}))
8	6, 7	mpan2 519	. . . 4 ⊢ (B ≼ A → B ≼ (A × {1o}))
9		0ex 1745	. . . . . . . 8 ⊢ ∅ ∈ V
10	2, 9	xpsnen 3339	. . . . . . 7 ⊢ (A × {∅}) ≈ A
11	2, 10	ensymi 3318	. . . . . 6 ⊢ A ≈ (A × {∅})
12		endom 3289	. . . . . 6 ⊢ (A ≈ (A × {∅}) → A ≼ (A × {∅}))
13	11, 12	ax-mp 6	. . . . 5 ⊢ A ≼ (A × {∅})
14		0ne1oOLD 3113	. . . . . . 7 ⊢ ¬ ∅ = 1o
15		xpsndisj 2655	. . . . . . 7 ⊢ (¬ ∅ = 1o → ((A × {∅}) ∩ (A × {1o})) = ∅)
16	14, 15	ax-mp 6	. . . . . 6 ⊢ ((A × {∅}) ∩ (A × {1o})) = ∅
17		p0ex 1885	. . . . . . . 8 ⊢ {∅} ∈ V
18	2, 17	xpex 2488	. . . . . . 7 ⊢ (A × {∅}) ∈ V
19		unxpdom2.2	. . . . . . 7 ⊢ B ∈ V
20		snex 1859	. . . . . . . 8 ⊢ {1o} ∈ V
21	2, 20	xpex 2488	. . . . . . 7 ⊢ (A × {1o}) ∈ V
22	18, 19, 21	undom 3342	. . . . . 6 ⊢ (((A ≼ (A × {∅}) ∧ B ≼ (A × {1o})) ∧ ((A × {∅}) ∩ (A × {1o})) = ∅) → (A ∪ B) ≼ ((A × {∅}) ∪ (A × {1o})))
23	16, 22	mpan2 519	. . . . 5 ⊢ ((A ≼ (A × {∅}) ∧ B ≼ (A × {1o})) → (A ∪ B) ≼ ((A × {∅}) ∪ (A × {1o})))
24	13, 23	mpan 518	. . . 4 ⊢ (B ≼ (A × {1o}) → (A ∪ B) ≼ ((A × {∅}) ∪ (A × {1o})))
25	8, 24	syl 12	. . 3 ⊢ (B ≼ A → (A ∪ B) ≼ ((A × {∅}) ∪ (A × {1o})))
26		unxpdom 3650	. . . . 5 ⊢ ((1o ≺ (A × {∅}) ∧ 1o ≺ (A × {1o})) → ((A × {∅}) ∪ (A × {1o})) ≼ ((A × {∅}) × (A × {1o})))
27		sdomentr 3371	. . . . . . 7 ⊢ ((A × {∅}) ∈ V → ((1o ≺ A ∧ A ≈ (A × {∅})) → 1o ≺ (A × {∅})))
28	18, 27	ax-mp 6	. . . . . 6 ⊢ ((1o ≺ A ∧ A ≈ (A × {∅})) → 1o ≺ (A × {∅}))
29	11, 28	mpan2 519	. . . . 5 ⊢ (1o ≺ A → 1o ≺ (A × {∅}))
30		sdomentr 3371	. . . . . . 7 ⊢ ((A × {1o}) ∈ V → ((1o ≺ A ∧ A ≈ (A × {1o})) → 1o ≺ (A × {1o})))
31	21, 30	ax-mp 6	. . . . . 6 ⊢ ((1o ≺ A ∧ A ≈ (A × {1o})) → 1o ≺ (A × {1o}))
32	6, 31	mpan2 519	. . . . 5 ⊢ (1o ≺ A → 1o ≺ (A × {1o}))
33	26, 29, 32	sylanc 361	. . . 4 ⊢ (1o ≺ A → ((A × {∅}) ∪ (A × {1o})) ≼ ((A × {∅}) × (A × {1o})))
34	18, 2, 21, 2	xpen 3383	. . . . . 6 ⊢ (((A × {∅}) ≈ A ∧ (A × {1o}) ≈ A) → ((A × {∅}) × (A × {1o})) ≈ (A × A))
35	10, 5, 34	mp2an 520	. . . . 5 ⊢ ((A × {∅}) × (A × {1o})) ≈ (A × A)
36		domentr 3326	. . . . 5 ⊢ ((((A × {∅}) ∪ (A × {1o})) ≼ ((A × {∅}) × (A × {1o})) ∧ ((A × {∅}) × (A × {1o})) ≈ (A × A)) → ((A × {∅}) ∪ (A × {1o})) ≼ (A × A))
37	35, 36	mpan2 519	. . . 4 ⊢ (((A × {∅}) ∪ (A × {1o})) ≼ ((A × {∅}) × (A × {1o})) → ((A × {∅}) ∪ (A × {1o})) ≼ (A × A))
38	33, 37	syl 12	. . 3 ⊢ (1o ≺ A → ((A × {∅}) ∪ (A × {1o})) ≼ (A × A))
39	1, 25, 38	syl2an 349	. 2 ⊢ ((B ≼ A ∧ 1o ≺ A) → (A ∪ B) ≼ (A × A))
40	39	ancoms 334	1 ⊢ ((1o ≺ A ∧ B ≼ A) → (A ∪ B) ≼ (A × A))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  {csn 1808   class class class wbr 2054  ωcom 2372   × cxp 2408  1_oc1o 3099   ≈ cen 3271   ≼ cdom 3272   ≺ csdm 3273

This theorem is referenced by:  infxpidmlem12 4944

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-2o 3105  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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