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Theorem omniwomnimkv 7122

Description: A set is omniscient if and only if it is weakly omniscient and Markov. The case

says that LPO

WLPO $/\$ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)

**Assertion**
Ref	Expression
omniwomnimkv	Omni WOmni $/\$ Markov

Proof of Theorem omniwomnimkv Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2732	. 2 Omni
2		simpl 108	. . 3 WOmni $/\$ Markov WOmni
3	2	elexd 2734	. 2 WOmni $/\$ Markov
4		1n0 6391	. . . . . . . . . . . . . . 15
5	4	nesymi 2380	. . . . . . . . . . . . . 14
6		eqeq1 2171	. . . . . . . . . . . . . 14
7	5, 6	mtbiri 665	. . . . . . . . . . . . 13
8	7	reximi 2561	. . . . . . . . . . . 12
9		rexnalim 2453	. . . . . . . . . . . 12
10	8, 9	syl 14	. . . . . . . . . . 11
11	10	orim1i 750	. . . . . . . . . 10 $\/$ $\/$
12	11	orcomd 719	. . . . . . . . 9 $\/$ $\/$
13		df-dc 825	. . . . . . . . 9 DECID $\/$
14	12, 13	sylibr 133	. . . . . . . 8 $\/$ DECID
15	14	adantl 275	. . . . . . 7 $/\$ $/\$ $\/$ DECID
16		simpr 109	. . . . . . . . 9 $/\$ $/\$ $\/$ $\/$
17	16	orcomd 719	. . . . . . . 8 $/\$ $/\$ $\/$ $\/$
18	17	ord 714	. . . . . . 7 $/\$ $/\$ $\/$
19	15, 18	jca 304	. . . . . 6 $/\$ $/\$ $\/$ DECID $/\$
20		simprl 521	. . . . . . . . 9 $/\$ $/\$ DECID $/\$ DECID
21	20, 13	sylib 121	. . . . . . . 8 $/\$ $/\$ DECID $/\$ $\/$
22		simprr 522	. . . . . . . . 9 $/\$ $/\$ DECID $/\$
23	22	orim2d 778	. . . . . . . 8 $/\$ $/\$ DECID $/\$ $\/$ $\/$
24	21, 23	mpd 13	. . . . . . 7 $/\$ $/\$ DECID $/\$ $\/$
25	24	orcomd 719	. . . . . 6 $/\$ $/\$ DECID $/\$ $\/$
26	19, 25	impbida 586	. . . . 5 $/\$ $\/$ DECID $/\$
27	26	pm5.74da 440	. . . 4 $\/$ DECID $/\$
28	27	albidv 1811	. . 3 $\/$ DECID $/\$
29		isomni 7091	. . 3 Omni $\/$
30		iswomni 7120	. . . . . 6 WOmni DECID
31		ismkv 7108	. . . . . 6 Markov
32	30, 31	anbi12d 465	. . . . 5 WOmni $/\$ Markov DECID $/\$
33		19.26 1468	. . . . 5 DECID $/\$ DECID $/\$
34	32, 33	bitr4di 197	. . . 4 WOmni $/\$ Markov DECID $/\$
35		jcab 593	. . . . 5 DECID $/\$ DECID $/\$
36	35	albii 1457	. . . 4 DECID $/\$ DECID $/\$
37	34, 36	bitr4di 197	. . 3 WOmni $/\$ Markov DECID $/\$
38	28, 29, 37	3bitr4d 219	. 2 Omni WOmni $/\$ Markov
39	1, 3, 38	pm5.21nii 694	1 Omni WOmni $/\$ Markov

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 DECID wdc 824

wal 1340

wceq 1342

wcel 2135

wral 2442

wrex 2443

cvv 2721

c0 3404

wf 5178

cfv 5182

c1o 6368

c2o 6369 Omnicomni 7089 Markovcmarkov 7106 WOmnicwomni 7118

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-nul 4102

This theorem depends on definitions: df-bi 116 df-dc 825 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-nul 3405 df-sn 3576 df-suc 4343 df-fn 5185 df-f 5186 df-1o 6375 df-omni 7090 df-markov 7107 df-womni 7119

This theorem is referenced by: lpowlpo 7123

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