Theorem infxpidmlem1 7446

Description: Lemma for infxpidm 7458. 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 4391	. . . 4 |- ((x ~<_ (A u. B) /\ (A u. B) ~<_ x) -> x ~~ (A u. B))
2		infxpidmlem1.1	. . . . . . 7 |- A e. V
3		ssun1 2164	. . . . . . 7 |- A (_ (A u. B)
4		ssdomg 4343	. . . . . . 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 4355	. . . . . 6 |- ((x ~~ A /\ A ~<_ (A u. B)) -> x ~<_ (A u. B))
7	5, 6	mpan2 693	. . . . 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 4356	. . . . 5 |- (((A u. B) ~<_ (A X. B) /\ (A X. B) ~~ x) -> (A u. B) ~<_ x)
10		unxpdom 4767	. . . . . . . 8 |- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))
11		sdomentr 4404	. . . . . . . . 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 4404	. . . . . . . . 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 4349	. . . . . . . 8 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (A X. B)) -> x ~~ (A X. B))
20	2, 13	xpex 3222	. . . . . . . . 9 |- (A X. B) e. V
21	20	ensym 4347	. . . . . . . 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 1788	. . . . . . . 8 |- x e. V
24	23, 2, 23, 13	xpen 4420	. . . . . . 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 4191	. . 3 |- 1o e. om
31	23	infsdomnn 4463	. . 3 |- ((om ~<_ x /\ 1o e. om) -> 1o ~< x)
32	30, 31	mpan2 693	. 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 1105  Vcvv 1786   u. cun 2016   (_ wss 2018   class class class wbr 2587  omcom 3094   X. cxp 3131  1oc1o 4066   ~~ cen 4302   ~<_ cdom 4303   ~< csdm 4304

This theorem is referenced by:  infxpidmlem12 7457

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-inf2 4549  ax-ac 4668

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-sbc 1913  df-csb 1973  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-pss 2026  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-rdg 3871  df-opr 3904  df-oprab 3905  df-1st 4017  df-2nd 4018  df-1o 4071  df-2o 4072  df-er 4199  df-en 4305  df-dom 4306  df-sdom 4307  df-card 4740

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