Theorem wecmpep 5106

Description: The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)

Assertion

Ref	Expression
wecmpep	|- ( ( _E We A /\ ( x e. A /\ y e. A ) ) -> ( x e. y \/ x = y \/ y e. x ) )

Proof of Theorem wecmpep

Step	Hyp	Ref	Expression
1		weso 5105	. 2 |- ( _E We A -> _E Or A )
2		solin 5058	. . 3 |- ( ( _E Or A /\ ( x e. A /\ y e. A ) ) -> ( x _E y \/ x = y \/ y _E x ) )
3		epel 5032	. . . 4 |- ( x _E y <-> x e. y )
4		biid 251	. . . 4 |- ( x = y <-> x = y )
5		epel 5032	. . . 4 |- ( y _E x <-> y e. x )
6	3, 4, 5	3orbi123i 1252	. . 3 |- ( ( x _E y \/ x = y \/ y _E x ) <-> ( x e. y \/ x = y \/ y e. x ) )
7	2, 6	sylib 208	. 2 |- ( ( _E Or A /\ ( x e. A /\ y e. A ) ) -> ( x e. y \/ x = y \/ y e. x ) )
8	1, 7	sylan 488	1 |- ( ( _E We A /\ ( x e. A /\ y e. A ) ) -> ( x e. y \/ x = y \/ y e. x ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   e. wcel 1990   class class class wbr 4653   _E cep 5028   Or wor 5034   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  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-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-eprel 5029  df-so 5036  df-we 5075

This theorem is referenced by:  tz7.7  5749