Theorem infxpidmlem1 7512

Description: Lemma for infxpidm 7524. 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 4446	. . . 4 ⊢ ((x ≼ (A ∪ B) ⋀ (A ∪ B) ≼ x) → x ≈ (A ∪ B))
2		infxpidmlem1.1	. . . . . . 7 ⊢ A ∈ V
3		ssun1 2190	. . . . . . 7 ⊢ A ⊆ (A ∪ B)
4		ssdomg 4398	. . . . . . 7 ⊢ (A ∈ V → (A ⊆ (A ∪ B) → A ≼ (A ∪ B)))
5	2, 3, 4	mp2 43	. . . . . 6 ⊢ A ≼ (A ∪ B)
6		endomtr 4410	. . . . . 6 ⊢ ((x ≈ A ⋀ A ≼ (A ∪ B)) → x ≼ (A ∪ B))
7	5, 6	mpan2 695	. . . . 5 ⊢ (x ≈ A → x ≼ (A ∪ B))
8	7	ad2antrl 406	. . . 4 ⊢ (((1o ≺ x ⋀ x ≈ (x × x)) ⋀ (x ≈ A ⋀ x ≈ B)) → x ≼ (A ∪ B))
9		domentr 4411	. . . . 5 ⊢ (((A ∪ B) ≼ (A × B) ⋀ (A × B) ≈ x) → (A ∪ B) ≼ x)
10		unxpdom 4827	. . . . . . . 8 ⊢ ((1o ≺ A ⋀ 1o ≺ B) → (A ∪ B) ≼ (A × B))
11		sdomentr 4459	. . . . . . . . 9 ⊢ (A ∈ V → ((1o ≺ x ⋀ x ≈ A) → 1o ≺ A))
12	2, 11	ax-mp 7	. . . . . . . 8 ⊢ ((1o ≺ x ⋀ x ≈ A) → 1o ≺ A)
13		infxpidmlem1.2	. . . . . . . . 9 ⊢ B ∈ V
14		sdomentr 4459	. . . . . . . . 9 ⊢ (B ∈ V → ((1o ≺ x ⋀ x ≈ B) → 1o ≺ B))
15	13, 14	ax-mp 7	. . . . . . . 8 ⊢ ((1o ≺ x ⋀ x ≈ B) → 1o ≺ B)
16	10, 12, 15	syl2an 454	. . . . . . 7 ⊢ (((1o ≺ x ⋀ x ≈ A) ⋀ (1o ≺ x ⋀ x ≈ B)) → (A ∪ B) ≼ (A × B))
17	16	anandis 512	. . . . . 6 ⊢ ((1o ≺ x ⋀ (x ≈ A ⋀ x ≈ B)) → (A ∪ B) ≼ (A × B))
18	17	adantlr 393	. . . . 5 ⊢ (((1o ≺ x ⋀ x ≈ (x × x)) ⋀ (x ≈ A ⋀ x ≈ B)) → (A ∪ B) ≼ (A × B))
19		entrt 4404	. . . . . . . 8 ⊢ ((x ≈ (x × x) ⋀ (x × x) ≈ (A × B)) → x ≈ (A × B))
20	2, 13	xpex 3256	. . . . . . . . 9 ⊢ (A × B) ∈ V
21	20	ensym 4402	. . . . . . . 8 ⊢ (x ≈ (A × B) → (A × B) ≈ x)
22	19, 21	syl 10	. . . . . . 7 ⊢ ((x ≈ (x × x) ⋀ (x × x) ≈ (A × B)) → (A × B) ≈ x)
23		visset 1810	. . . . . . . 8 ⊢ x ∈ V
24	23, 2, 23, 13	xpen 4477	. . . . . . 7 ⊢ ((x ≈ A ⋀ x ≈ B) → (x × x) ≈ (A × B))
25	22, 24	sylan2 451	. . . . . 6 ⊢ ((x ≈ (x × x) ⋀ (x ≈ A ⋀ x ≈ B)) → (A × B) ≈ x)
26	25	adantll 392	. . . . 5 ⊢ (((1o ≺ x ⋀ x ≈ (x × x)) ⋀ (x ≈ A ⋀ x ≈ B)) → (A × B) ≈ x)
27	9, 18, 26	sylanc 471	. . . 4 ⊢ (((1o ≺ x ⋀ x ≈ (x × x)) ⋀ (x ≈ A ⋀ x ≈ B)) → (A ∪ B) ≼ x)
28	1, 8, 27	sylanc 471	. . 3 ⊢ (((1o ≺ x ⋀ x ≈ (x × x)) ⋀ (x ≈ A ⋀ x ≈ B)) → x ≈ (A ∪ B))
29	28	ex 373	. 2 ⊢ ((1o ≺ x ⋀ x ≈ (x × x)) → ((x ≈ A ⋀ x ≈ B) → x ≈ (A ∪ B)))
30		1onn 4246	. . 3 ⊢ 1o ∈ ω
31	23	infsdomnn 4520	. . 3 ⊢ ((ω ≼ x ⋀ 1o ∈ ω) → 1o ≺ x)
32	30, 31	mpan2 695	. 2 ⊢ (ω ≼ x → 1o ≺ x)
33	29, 32	sylan 448	1 ⊢ ((ω ≼ x ⋀ x ≈ (x × x)) → ((x ≈ A ⋀ x ≈ B) → x ≈ (A ∪ B)))

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 957  Vcvv 1808   ∪ cun 2042   ⊆ wss 2044   class class class wbr 2615  ωcom 3127   × cxp 3164  1_oc1o 4121   ≈ cen 4357   ≼ cdom 4358   ≺ csdm 4359

This theorem is referenced by:  infxpidmlem12 7523

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-2o 4127  df-er 4254  df-en 4360  df-dom 4361  df-sdom 4362  df-card 4799

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