Theorem unxpdom2 5997

Description: Corollary of unxpdom 5996.

Hypotheses

Ref	Expression
unxpdom2.1	|- A e. _V
unxpdom2.2	|- B e. _V

Assertion

Ref	Expression
unxpdom2	|- ((1o ~< A /\ B ~<_ A) -> (A u. B) ~<_ (A X. A))

Proof of Theorem unxpdom2

Step	Hyp	Ref	Expression
1		domtr 5474	. . 3 |- (((A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})) /\ ((A X. {1o}) u. (A X. {(/)})) ~<_ (A X. A)) -> (A u. B) ~<_ (A X. A))
2		1n0 5187	. . . . . 6 |- 1o =/= (/)
3		xpsndisj 4339	. . . . . 6 |- (1o =/= (/) -> ((A X. {1o}) i^i (A X. {(/)})) = (/))
4	2, 3	ax-mp 7	. . . . 5 |- ((A X. {1o}) i^i (A X. {(/)})) = (/)
5		unxpdom2.1	. . . . . . 7 |- A e. _V
6		snex 3492	. . . . . . 7 |- {1o} e. _V
7	5, 6	xpex 4096	. . . . . 6 |- (A X. {1o}) e. _V
8		unxpdom2.2	. . . . . 6 |- B e. _V
9		p0ex 3495	. . . . . . 7 |- {(/)} e. _V
10	5, 9	xpex 4096	. . . . . 6 |- (A X. {(/)}) e. _V
11	7, 8, 10	undom 5497	. . . . 5 |- (((A ~<_ (A X. {1o}) /\ B ~<_ (A X. {(/)})) /\ ((A X. {1o}) i^i (A X. {(/)})) = (/)) -> (A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})))
12	4, 11	mpan2 760	. . . 4 |- ((A ~<_ (A X. {1o}) /\ B ~<_ (A X. {(/)})) -> (A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})))
13		1onn 5310	. . . . . . . 8 |- 1o e. om
14	13	elisseti 2301	. . . . . . 7 |- 1o e. _V
15	5, 14	xpsnen 5494	. . . . . 6 |- (A X. {1o}) ~~ A
16	5, 15	ensymi 5472	. . . . 5 |- A ~~ (A X. {1o})
17		endom 5444	. . . . 5 |- (A ~~ (A X. {1o}) -> A ~<_ (A X. {1o}))
18	16, 17	ax-mp 7	. . . 4 |- A ~<_ (A X. {1o})
19		0ex 3446	. . . . . . 7 |- (/) e. _V
20	5, 19	xpsnen 5494	. . . . . 6 |- (A X. {(/)}) ~~ A
21	5, 20	ensymi 5472	. . . . 5 |- A ~~ (A X. {(/)})
22		domentr 5480	. . . . 5 |- ((B ~<_ A /\ A ~~ (A X. {(/)})) -> B ~<_ (A X. {(/)}))
23	21, 22	mpan2 760	. . . 4 |- (B ~<_ A -> B ~<_ (A X. {(/)}))
24	12, 18, 23	sylancr 526	. . 3 |- (B ~<_ A -> (A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})))
25		domentr 5480	. . . 4 |- ((((A X. {1o}) u. (A X. {(/)})) ~<_ ((A X. {1o}) X. (A X. {(/)})) /\ ((A X. {1o}) X. (A X. {(/)})) ~~ (A X. A)) -> ((A X. {1o}) u. (A X. {(/)})) ~<_ (A X. A))
26		sdomentr 5533	. . . . . . 7 |- ((A X. {1o}) e. _V -> ((1o ~< A /\ A ~~ (A X. {1o})) -> 1o ~< (A X. {1o})))
27	7, 26	ax-mp 7	. . . . . 6 |- ((1o ~< A /\ A ~~ (A X. {1o})) -> 1o ~< (A X. {1o}))
28	16, 27	mpan2 760	. . . . 5 |- (1o ~< A -> 1o ~< (A X. {1o}))
29		sdomentr 5533	. . . . . . 7 |- ((A X. {(/)}) e. _V -> ((1o ~< A /\ A ~~ (A X. {(/)})) -> 1o ~< (A X. {(/)})))
30	10, 29	ax-mp 7	. . . . . 6 |- ((1o ~< A /\ A ~~ (A X. {(/)})) -> 1o ~< (A X. {(/)}))
31	21, 30	mpan2 760	. . . . 5 |- (1o ~< A -> 1o ~< (A X. {(/)}))
32		unxpdom 5996	. . . . 5 |- ((1o ~< (A X. {1o}) /\ 1o ~< (A X. {(/)})) -> ((A X. {1o}) u. (A X. {(/)})) ~<_ ((A X. {1o}) X. (A X. {(/)})))
33	28, 31, 32	syl11anc 524	. . . 4 |- (1o ~< A -> ((A X. {1o}) u. (A X. {(/)})) ~<_ ((A X. {1o}) X. (A X. {(/)})))
34	7, 5, 10, 5	xpen 5582	. . . . 5 |- (((A X. {1o}) ~~ A /\ (A X. {(/)}) ~~ A) -> ((A X. {1o}) X. (A X. {(/)})) ~~ (A X. A))
35	15, 20, 34	mp2an 761	. . . 4 |- ((A X. {1o}) X. (A X. {(/)})) ~~ (A X. A)
36	25, 33, 35	sylancl 525	. . 3 |- (1o ~< A -> ((A X. {1o}) u. (A X. {(/)})) ~<_ (A X. A))
37	1, 24, 36	syl2an 503	. 2 |- ((B ~<_ A /\ 1o ~< A) -> (A u. B) ~<_ (A X. A))
38	37	ancoms 484	1 |- ((1o ~< A /\ B ~<_ A) -> (A u. B) ~<_ (A X. A))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044   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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