Theorem welb 33531

Description: A nonempty subset of a well-ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.)

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 `' R y /\ A. y e. B ( y `' R x -> E. z e. B y `' R z ) ) ) )

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 5101	. . . . . 6 |- ( B C_ A -> ( R We A -> R We B ) )
2	1	impcom 446	. . . . 5 |- ( ( R We A /\ B C_ A ) -> R We B )
3		weso 5105	. . . . 5 |- ( R We B -> R Or B )
4	2, 3	syl 17	. . . 4 |- ( ( R We A /\ B C_ A ) -> R Or B )
5		cnvso 5674	. . . 4 |- ( R Or B <-> `' R Or B )
6	4, 5	sylib 208	. . 3 |- ( ( R We A /\ B C_ A ) -> `' R Or B )
7	6	3ad2antr2 1227	. 2 |- ( ( R We A /\ ( B e. C /\ B C_ A /\ B =/= (/) ) ) -> `' R Or B )
8		wefr 5104	. . . . 5 |- ( R We B -> R Fr B )
9	2, 8	syl 17	. . . 4 |- ( ( R We A /\ B C_ A ) -> R Fr B )
10	9	3ad2antr2 1227	. . 3 |- ( ( R We A /\ ( B e. C /\ B C_ A /\ B =/= (/) ) ) -> R Fr B )
11		ssid 3624	. . . . . 6 |- B C_ B
12	11	a1i 11	. . . . 5 |- ( B C_ A -> B C_ B )
13	12	3anim2i 1249	. . . 4 |- ( ( B e. C /\ B C_ A /\ B =/= (/) ) -> ( B e. C /\ B C_ B /\ B =/= (/) ) )
14	13	adantl 482	. . 3 |- ( ( R We A /\ ( B e. C /\ B C_ A /\ B =/= (/) ) ) -> ( B e. C /\ B C_ B /\ B =/= (/) ) )
15		frinfm 33530	. . 3 |- ( ( R Fr B /\ ( B e. C /\ B C_ B /\ B =/= (/) ) ) -> E. x e. B ( A. y e. B -. x `' R y /\ A. y e. B ( y `' R x -> E. z e. B y `' R z ) ) )
16	10, 14, 15	syl2anc 693	. 2 |- ( ( R We A /\ ( B e. C /\ B C_ A /\ B =/= (/) ) ) -> E. x e. B ( A. y e. B -. x `' R y /\ A. y e. B ( y `' R x -> E. z e. B y `' R z ) ) )
17	7, 16	jca 554	1 |- ( ( R We A /\ ( B e. C /\ B C_ A /\ B =/= (/) ) ) -> ( `' R Or B /\ E. x e. B ( A. y e. B -. x `' R y /\ A. y e. B ( y `' R x -> E. z e. B y `' R z ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Or wor 5034   Fr wfr 5070   We wwe 5072   `'ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-cnv 5122

This theorem is referenced by: (None)