Theorem infxpidmlem1 8821

Description: Lemma for infxpidm 8833. 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		infxpidmlem1.1	. . . . . . 7 |- A e. _V
2		ssun1 2767	. . . . . . 7 |- A C_ (A u. B)
3		ssdomg 5467	. . . . . . 7 |- (A e. _V -> (A C_ (A u. B) -> A ~<_ (A u. B)))
4	1, 2, 3	mp2 54	. . . . . 6 |- A ~<_ (A u. B)
5		endomtr 5479	. . . . . 6 |- ((x ~~ A /\ A ~<_ (A u. B)) -> x ~<_ (A u. B))
6	4, 5	mpan2 760	. . . . 5 |- (x ~~ A -> x ~<_ (A u. B))
7	6	ad2antrl 442	. . . 4 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> x ~<_ (A u. B))
8		unxpdom 5996	. . . . . . . 8 |- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))
9		sdomentr 5533	. . . . . . . . 9 |- (A e. _V -> ((1o ~< x /\ x ~~ A) -> 1o ~< A))
10	1, 9	ax-mp 7	. . . . . . . 8 |- ((1o ~< x /\ x ~~ A) -> 1o ~< A)
11		infxpidmlem1.2	. . . . . . . . 9 |- B e. _V
12		sdomentr 5533	. . . . . . . . 9 |- (B e. _V -> ((1o ~< x /\ x ~~ B) -> 1o ~< B))
13	11, 12	ax-mp 7	. . . . . . . 8 |- ((1o ~< x /\ x ~~ B) -> 1o ~< B)
14	8, 10, 13	syl2an 503	. . . . . . 7 |- (((1o ~< x /\ x ~~ A) /\ (1o ~< x /\ x ~~ B)) -> (A u. B) ~<_ (A X. B))
15	14	anandis 570	. . . . . 6 |- ((1o ~< x /\ (x ~~ A /\ x ~~ B)) -> (A u. B) ~<_ (A X. B))
16	15	adantlr 429	. . . . 5 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> (A u. B) ~<_ (A X. B))
17		entr 5473	. . . . . . . 8 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (A X. B)) -> x ~~ (A X. B))
18	1, 11	xpex 4096	. . . . . . . . 9 |- (A X. B) e. _V
19	18	ensym 5471	. . . . . . . 8 |- (x ~~ (A X. B) -> (A X. B) ~~ x)
20	17, 19	syl 12	. . . . . . 7 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (A X. B)) -> (A X. B) ~~ x)
21		visset 2295	. . . . . . . 8 |- x e. _V
22	21, 1, 21, 11	xpen 5582	. . . . . . 7 |- ((x ~~ A /\ x ~~ B) -> (x X. x) ~~ (A X. B))
23	20, 22	sylan2 500	. . . . . 6 |- ((x ~~ (x X. x) /\ (x ~~ A /\ x ~~ B)) -> (A X. B) ~~ x)
24	23	adantll 428	. . . . 5 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> (A X. B) ~~ x)
25		domentr 5480	. . . . 5 |- (((A u. B) ~<_ (A X. B) /\ (A X. B) ~~ x) -> (A u. B) ~<_ x)
26	16, 24, 25	syl11anc 524	. . . 4 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> (A u. B) ~<_ x)
27		sbth 5520	. . . 4 |- ((x ~<_ (A u. B) /\ (A u. B) ~<_ x) -> x ~~ (A u. B))
28	7, 26, 27	syl11anc 524	. . 3 |- (((1o ~< x /\ x ~~ (x X. x)) /\ (x ~~ A /\ x ~~ B)) -> x ~~ (A u. B))
29	28	ex 402	. 2 |- ((1o ~< x /\ x ~~ (x X. x)) -> ((x ~~ A /\ x ~~ B) -> x ~~ (A u. B)))
30		1onn 5310	. . 3 |- 1o e. om
31	21	infsdomnn 5625	. . 3 |- ((om ~<_ x /\ 1o e. om) -> 1o ~< x)
32	30, 31	mpan2 760	. 2 |- (om ~<_ x -> 1o ~< x)
33	29, 32	sylan 497	1 |- ((om ~<_ x /\ x ~~ (x X. x)) -> ((x ~~ A /\ x ~~ B) -> x ~~ (A u. B)))

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  _Vcvv 2292   u. cun 2591   C_ wss 2593   class class class wbr 3338  omcom 3949   X. cxp 3984  1oc1o 5172   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425

This theorem is referenced by:  infxpidmlem12 8832

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-2o 5178  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862

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