Theorem infxpidmlem1 4933

Description: Lemma for infxpidm 4945. An infinite idempotent set x is equinumerous to the union of any two sets A and B equinumerous to it.

Hypotheses

Ref	Expression
infxpidmlem1.1	⊢ A ∈ V
infxpidmlem1.2	⊢ B ∈ V

Assertion

Ref	Expression
infxpidmlem1	⊢ ((ω ≼ x ∧ x ≈ (x × x)) → ((x ≈ A ∧ x ≈ B) → x ≈ (A ∪ B)))

Proof of Theorem infxpidmlem1

Step	Hyp	Ref	Expression
1		sbth 3359	. . . 4 ⊢ ((x ≼ (A ∪ B) ∧ (A ∪ B) ≼ x) → x ≈ (A ∪ B))
2		infxpidmlem1.1	. . . . . . 7 ⊢ A ∈ V
3		ssun1 1621	. . . . . . 7 ⊢ A ⊆ (A ∪ B)
4		ssdomg 3311	. . . . . . 7 ⊢ (A ∈ V → (A ⊆ (A ∪ B) → A ≼ (A ∪ B)))
5	2, 3, 4	mp2 43	. . . . . 6 ⊢ A ≼ (A ∪ B)
6		endomtr 3325	. . . . . 6 ⊢ ((x ≈ A ∧ A ≼ (A ∪ B)) → x ≼ (A ∪ B))
7	5, 6	mpan2 519	. . . . 5 ⊢ (x ≈ A → x ≼ (A ∪ B))
8	7	ad2antrl 322	. . . 4 ⊢ (((1o ≺ x ∧ x ≈ (x × x)) ∧ (x ≈ A ∧ x ≈ B)) → x ≼ (A ∪ B))
9		domentr 3326	. . . . 5 ⊢ (((A ∪ B) ≼ (A × B) ∧ (A × B) ≈ x) → (A ∪ B) ≼ x)
10		unxpdom 3650	. . . . . . . 8 ⊢ ((1o ≺ A ∧ 1o ≺ B) → (A ∪ B) ≼ (A × B))
11		sdomentr 3371	. . . . . . . . 9 ⊢ (A ∈ V → ((1o ≺ x ∧ x ≈ A) → 1o ≺ A))
12	2, 11	ax-mp 6	. . . . . . . 8 ⊢ ((1o ≺ x ∧ x ≈ A) → 1o ≺ A)
13		infxpidmlem1.2	. . . . . . . . 9 ⊢ B ∈ V
14		sdomentr 3371	. . . . . . . . 9 ⊢ (B ∈ V → ((1o ≺ x ∧ x ≈ B) → 1o ≺ B))
15	13, 14	ax-mp 6	. . . . . . . 8 ⊢ ((1o ≺ x ∧ x ≈ B) → 1o ≺ B)
16	10, 12, 15	syl2an 349	. . . . . . 7 ⊢ (((1o ≺ x ∧ x ≈ A) ∧ (1o ≺ x ∧ x ≈ B)) → (A ∪ B) ≼ (A × B))
17	16	anandis 394	. . . . . 6 ⊢ ((1o ≺ x ∧ (x ≈ A ∧ x ≈ B)) → (A ∪ B) ≼ (A × B))
18	17	adantlr 310	. . . . 5 ⊢ (((1o ≺ x ∧ x ≈ (x × x)) ∧ (x ≈ A ∧ x ≈ B)) → (A ∪ B) ≼ (A × B))
19		entrt 3319	. . . . . . . 8 ⊢ ((x ≈ (x × x) ∧ (x × x) ≈ (A × B)) → x ≈ (A × B))
20	2, 13	xpex 2488	. . . . . . . . 9 ⊢ (A × B) ∈ V
21	20	ensym 3317	. . . . . . . 8 ⊢ (x ≈ (A × B) → (A × B) ≈ x)
22	19, 21	syl 12	. . . . . . 7 ⊢ ((x ≈ (x × x) ∧ (x × x) ≈ (A × B)) → (A × B) ≈ x)
23		visset 1350	. . . . . . . 8 ⊢ x ∈ V
24	23, 2, 23, 13	xpen 3383	. . . . . . 7 ⊢ ((x ≈ A ∧ x ≈ B) → (x × x) ≈ (A × B))
25	22, 24	sylan2 346	. . . . . 6 ⊢ ((x ≈ (x × x) ∧ (x ≈ A ∧ x ≈ B)) → (A × B) ≈ x)
26	25	adantll 309	. . . . 5 ⊢ (((1o ≺ x ∧ x ≈ (x × x)) ∧ (x ≈ A ∧ x ≈ B)) → (A × B) ≈ x)
27	9, 18, 26	sylanc 361	. . . 4 ⊢ (((1o ≺ x ∧ x ≈ (x × x)) ∧ (x ≈ A ∧ x ≈ B)) → (A ∪ B) ≼ x)
28	1, 8, 27	sylanc 361	. . 3 ⊢ (((1o ≺ x ∧ x ≈ (x × x)) ∧ (x ≈ A ∧ x ≈ B)) → x ≈ (A ∪ B))
29	28	exp 291	. 2 ⊢ ((1o ≺ x ∧ x ≈ (x × x)) → ((x ≈ A ∧ x ≈ B) → x ≈ (A ∪ B)))
30		1onn 3193	. . 3 ⊢ 1o ∈ ω
31	23	infsdomnn 3426	. . 3 ⊢ ((ω ≼ x ∧ 1o ∈ ω) → 1o ≺ x)
32	30, 31	mpan2 519	. 2 ⊢ (ω ≼ x → 1o ≺ x)
33	29, 32	sylan 343	1 ⊢ ((ω ≼ x ∧ x ≈ (x × x)) → ((x ≈ A ∧ x ≈ B) → x ≈ (A ∪ B)))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ⊆ wss 1487   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

metamath.org