Theorem unxpdom2 4817

Description: Corollary of unxpdom 4816.

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 4396	. . 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		unxpdom2.1	. . . . . 6 |- A e. V
3		0ex 2701	. . . . . . 7 |- (/) e. V
4	2, 3	xpsnen 4415	. . . . . 6 |- (A X. {(/)}) ~~ A
5	2, 4	ensymi 4394	. . . . 5 |- A ~~ (A X. {(/)})
6		domentr 4402	. . . . 5 |- ((B ~<_ A /\ A ~~ (A X. {(/)})) -> B ~<_ (A X. {(/)}))
7	5, 6	mpan2 694	. . . 4 |- (B ~<_ A -> B ~<_ (A X. {(/)}))
8		1onn 4237	. . . . . . . . 9 |- 1o e. om
9	8	elisseti 1809	. . . . . . . 8 |- 1o e. V
10	2, 9	xpsnen 4415	. . . . . . 7 |- (A X. {1o}) ~~ A
11	2, 10	ensymi 4394	. . . . . 6 |- A ~~ (A X. {1o})
12		endom 4366	. . . . . 6 |- (A ~~ (A X. {1o}) -> A ~<_ (A X. {1o}))
13	11, 12	ax-mp 7	. . . . 5 |- A ~<_ (A X. {1o})
14		1ne0 4126	. . . . . . 7 |- 1o =/= (/)
15		xpsndisj 3456	. . . . . . 7 |- (1o =/= (/) -> ((A X. {1o}) i^i (A X. {(/)})) = (/))
16	14, 15	ax-mp 7	. . . . . 6 |- ((A X. {1o}) i^i (A X. {(/)})) = (/)
17		snex 2740	. . . . . . . 8 |- {1o} e. V
18	2, 17	xpex 3250	. . . . . . 7 |- (A X. {1o}) e. V
19		unxpdom2.2	. . . . . . 7 |- B e. V
20		p0ex 2760	. . . . . . . 8 |- {(/)} e. V
21	2, 20	xpex 3250	. . . . . . 7 |- (A X. {(/)}) e. V
22	18, 19, 21	undom 4418	. . . . . 6 |- (((A ~<_ (A X. {1o}) /\ B ~<_ (A X. {(/)})) /\ ((A X. {1o}) i^i (A X. {(/)})) = (/)) -> (A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})))
23	16, 22	mpan2 694	. . . . 5 |- ((A ~<_ (A X. {1o}) /\ B ~<_ (A X. {(/)})) -> (A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})))
24	13, 23	mpan 693	. . . 4 |- (B ~<_ (A X. {(/)}) -> (A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})))
25	7, 24	syl 10	. . 3 |- (B ~<_ A -> (A u. B) ~<_ ((A X. {1o}) u. (A X. {(/)})))
26		unxpdom 4816	. . . . 5 |- ((1o ~< (A X. {1o}) /\ 1o ~< (A X. {(/)})) -> ((A X. {1o}) u. (A X. {(/)})) ~<_ ((A X. {1o}) X. (A X. {(/)})))
27		sdomentr 4450	. . . . . . 7 |- ((A X. {1o}) e. V -> ((1o ~< A /\ A ~~ (A X. {1o})) -> 1o ~< (A X. {1o})))
28	18, 27	ax-mp 7	. . . . . 6 |- ((1o ~< A /\ A ~~ (A X. {1o})) -> 1o ~< (A X. {1o}))
29	11, 28	mpan2 694	. . . . 5 |- (1o ~< A -> 1o ~< (A X. {1o}))
30		sdomentr 4450	. . . . . . 7 |- ((A X. {(/)}) e. V -> ((1o ~< A /\ A ~~ (A X. {(/)})) -> 1o ~< (A X. {(/)})))
31	21, 30	ax-mp 7	. . . . . 6 |- ((1o ~< A /\ A ~~ (A X. {(/)})) -> 1o ~< (A X. {(/)}))
32	5, 31	mpan2 694	. . . . 5 |- (1o ~< A -> 1o ~< (A X. {(/)}))
33	26, 29, 32	sylanc 471	. . . 4 |- (1o ~< A -> ((A X. {1o}) u. (A X. {(/)})) ~<_ ((A X. {1o}) X. (A X. {(/)})))
34	18, 2, 21, 2	xpen 4468	. . . . . 6 |- (((A X. {1o}) ~~ A /\ (A X. {(/)}) ~~ A) -> ((A X. {1o}) X. (A X. {(/)})) ~~ (A X. A))
35	10, 4, 34	mp2an 695	. . . . 5 |- ((A X. {1o}) X. (A X. {(/)})) ~~ (A X. A)
36		domentr 4402	. . . . 5 |- ((((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))
37	35, 36	mpan2 694	. . . 4 |- (((A X. {1o}) u. (A X. {(/)})) ~<_ ((A X. {1o}) X. (A X. {(/)})) -> ((A X. {1o}) u. (A X. {(/)})) ~<_ (A X. A))
38	33, 37	syl 10	. . 3 |- (1o ~< A -> ((A X. {1o}) u. (A X. {(/)})) ~<_ (A X. A))
39	1, 25, 38	syl2an 454	. 2 |- ((B ~<_ A /\ 1o ~< A) -> (A u. B) ~<_ (A X. A))
40	39	ancoms 436	1 |- ((1o ~< A /\ B ~<_ A) -> (A u. B) ~<_ (A X. A))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   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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