Theorem welb 15759

Description: A non-empty subset of a well ordered set has a lower bound.

Assertion

Ref	Expression
welb	|- ((R We A /\ (B e. C /\ B C_ A /\ B =/= (/))) -> (`'R Or B /\ E.x e. B (A.y e. B -. x`'Ry /\ A.y e. B (y`'Rx -> E.z e. B y`'Rz))))

Distinct variable groups:   x,A,y,z   x,B,y,z   x,C,y,z   x,R,y,z

Proof of Theorem welb

Step	Hyp	Ref	Expression
1		wess 3645	. . . . 5 |- (B C_ A -> (R We A -> R We B))
2	1	impcom 378	. . . 4 |- ((R We A /\ B C_ A) -> R We B)
3		weso 3649	. . . 4 |- (R We B -> R Or B)
4		cnvso 4428	. . . . 5 |- (R Or B <-> `'R Or B)
5	4	biimpi 168	. . . 4 |- (R Or B -> `'R Or B)
6	2, 3, 5	3syl 24	. . 3 |- ((R We A /\ B C_ A) -> `'R Or B)
7	6	3ad2antr2 1042	. 2 |- ((R We A /\ (B e. C /\ B C_ A /\ B =/= (/))) -> `'R Or B)
8		wefr 3648	. . . . 5 |- (R We B -> R Fr B)
9	2, 8	syl 12	. . . 4 |- ((R We A /\ B C_ A) -> R Fr B)
10	9	3ad2antr2 1042	. . 3 |- ((R We A /\ (B e. C /\ B C_ A /\ B =/= (/))) -> R Fr B)
11		id 73	. . . . 5 |- (B e. C -> B e. C)
12		ssid 2634	. . . . . 6 |- B C_ B
13	12	a1i 8	. . . . 5 |- (B C_ A -> B C_ B)
14		id 73	. . . . 5 |- (B =/= (/) -> B =/= (/))
15	11, 13, 14	3anim123i 1053	. . . 4 |- ((B e. C /\ B C_ A /\ B =/= (/)) -> (B e. C /\ B C_ B /\ B =/= (/)))
16	15	adantl 424	. . 3 |- ((R We A /\ (B e. C /\ B C_ A /\ B =/= (/))) -> (B e. C /\ B C_ B /\ B =/= (/)))
17		frinfm 15758	. . 3 |- ((R Fr B /\ (B e. C /\ B C_ B /\ B =/= (/))) -> E.x e. B (A.y e. B -. x`'Ry /\ A.y e. B (y`'Rx -> E.z e. B y`'Rz)))
18	10, 16, 17	syl11anc 524	. 2 |- ((R We A /\ (B e. C /\ B C_ A /\ B =/= (/))) -> E.x e. B (A.y e. B -. x`'Ry /\ A.y e. B (y`'Rx -> E.z e. B y`'Rz)))
19	7, 18	jca 310	1 |- ((R We A /\ (B e. C /\ B C_ A /\ B =/= (/))) -> (`'R Or B /\ E.x e. B (A.y e. B -. x`'Ry /\ A.y e. B (y`'Rx -> E.z e. B y`'Rz))))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875   class class class wbr 3338   Or wor 3590   Fr wfr 3623   We wwe 3624  `'ccnv 3985

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-cnv 4002

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