Theorem wereu 5110

Description: A subset of a well-ordered set has a unique minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
wereu	|- ( ( R We A /\ ( B e. V /\ B C_ A /\ B =/= (/) ) ) -> E! x e. B A. y e. B -. y R x )

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

Allowed substitution hints:   V( x, y)

Proof of Theorem wereu

Step	Hyp	Ref	Expression
1		wefr 5104	. . 3 |- ( R We A -> R Fr A )
2		fri 5076	. . . . . 6 |- ( ( ( B e. V /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x )
3	2	exp32 631	. . . . 5 |- ( ( B e. V /\ R Fr A ) -> ( B C_ A -> ( B =/= (/) -> E. x e. B A. y e. B -. y R x ) ) )
4	3	expcom 451	. . . 4 |- ( R Fr A -> ( B e. V -> ( B C_ A -> ( B =/= (/) -> E. x e. B A. y e. B -. y R x ) ) ) )
5	4	3imp2 1282	. . 3 |- ( ( R Fr A /\ ( B e. V /\ B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x )
6	1, 5	sylan 488	. 2 |- ( ( R We A /\ ( B e. V /\ B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x )
7		weso 5105	. . . . 5 |- ( R We A -> R Or A )
8		soss 5053	. . . . 5 |- ( B C_ A -> ( R Or A -> R Or B ) )
9	7, 8	mpan9 486	. . . 4 |- ( ( R We A /\ B C_ A ) -> R Or B )
10		somo 5069	. . . 4 |- ( R Or B -> E* x e. B A. y e. B -. y R x )
11	9, 10	syl 17	. . 3 |- ( ( R We A /\ B C_ A ) -> E* x e. B A. y e. B -. y R x )
12	11	3ad2antr2 1227	. 2 |- ( ( R We A /\ ( B e. V /\ B C_ A /\ B =/= (/) ) ) -> E* x e. B A. y e. B -. y R x )
13		reu5 3159	. 2 |- ( E! x e. B A. y e. B -. y R x <-> ( E. x e. B A. y e. B -. y R x /\ E* x e. B A. y e. B -. y R x ) )
14	6, 12, 13	sylanbrc 698	1 |- ( ( R We A /\ ( B e. V /\ B C_ A /\ B =/= (/) ) ) -> E! x e. B A. y e. B -. y R x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   E*wrmo 2915   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Or wor 5034   Fr wfr 5070   We wwe 5072

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

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-reu 2919  df-rmo 2920  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-po 5035  df-so 5036  df-fr 5073  df-we 5075

This theorem is referenced by:  htalem  8759  zorn2lem1  9318  dyadmax  23366  wessf1ornlem  39371