Theorem infxpidmlem1 7495

Description: Lemma for infxpidm 7507. 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 e. V
infxpidmlem1.2	|- B e. V

Assertion

Ref	Expression
infxpidmlem1	|- ((om ~<_ x /\ x ~~ (x X. x)) -> ((x ~~ A /\ x ~~ B) -> x ~~ (A u. B)))

Proof of Theorem infxpidmlem1

Step	Hyp	Ref	Expression
1		sbth 4437	. . . 4 |- ((x ~<_ (A u. B) /\ (A u. B) ~<_ x) -> x ~~ (A u. B))
2		infxpidmlem1.1	. . . . . . 7 |- A e. V
3		ssun1 2183	. . . . . . 7 |- A (_ (A u. B)
4		ssdomg 4389	. . . . . . 7 |- (A e. V -> (A (_ (A u. B) -> A ~<_ (A u. B)))
5	2, 3, 4	mp2 43	. . . . . 6 |- A ~<_ (A u. B)
6		endomtr 4401	. . . . . 6 |- ((x ~~ A /\ A ~<_ (A u. B)) -> x ~<_ (A u. B))
7	5, 6	mpan2 694	. . . . 5 |- (x ~~ A -> x ~<_ (A u. B))
8	7	ad2antrl 406	. . . 4 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> x ~<_ (A u. B))
9		domentr 4402	. . . . 5 |- (((A u. B) ~<_ (A X. B) /\ (A X. B) ~~ x) -> (A u. B) ~<_ x)
10		unxpdom 4816	. . . . . . . 8 |- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))
11		sdomentr 4450	. . . . . . . . 9 |- (A e. 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 e. V
14		sdomentr 4450	. . . . . . . . 9 |- (B e. 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 u. B) ~<_ (A X. B))
17	16	anandis 511	. . . . . 6 |- ((1o ~< x /\ (x ~~ A /\ x ~~ B)) -> (A u. B) ~<_ (A X. B))
18	17	adantlr 393	. . . . 5 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> (A u. B) ~<_ (A X. B))
19		entrt 4395	. . . . . . . 8 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (A X. B)) -> x ~~ (A X. B))
20	2, 13	xpex 3250	. . . . . . . . 9 |- (A X. B) e. V
21	20	ensym 4393	. . . . . . . 8 |- (x ~~ (A X. B) -> (A X. B) ~~ x)
22	19, 21	syl 10	. . . . . . 7 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (A X. B)) -> (A X. B) ~~ x)
23		visset 1804	. . . . . . . 8 |- x e. V
24	23, 2, 23, 13	xpen 4468	. . . . . . 7 |- ((x ~~ A /\ x ~~ B) -> (x X. x) ~~ (A X. B))
25	22, 24	sylan2 451	. . . . . 6 |- ((x ~~ (x X. x) /\ (x ~~ A /\ x ~~ B)) -> (A X. B) ~~ x)
26	25	adantll 392	. . . . 5 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> (A X. B) ~~ x)
27	9, 18, 26	sylanc 471	. . . 4 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> (A u. B) ~<_ x)
28	1, 8, 27	sylanc 471	. . 3 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> x ~~ (A u. B))
29	28	ex 373	. 2 |- ((1o ~< x /\ x ~~ (x X. x)) -> ((x ~~ A /\ x ~~ B) -> x ~~ (A u. B)))
30		1onn 4237	. . 3 |- 1o e. om
31	23	infsdomnn 4511	. . 3 |- ((om ~<_ x /\ 1o e. om) -> 1o ~< x)
32	30, 31	mpan2 694	. 2 |- (om ~<_ x -> 1o ~< x)
33	29, 32	sylan 448	1 |- ((om ~<_ x /\ x ~~ (x X. x)) -> ((x ~~ A /\ x ~~ B) -> x ~~ (A u. B)))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  Vcvv 1802   u. cun 2035   (_ wss 2037   class class class wbr 2609  omcom 3121   X. cxp 3158  1oc1o 4112   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350

This theorem is referenced by:  infxpidmlem12 7506

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-2o 4118  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788

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