Theorem bdrmo 13043

Description: Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdrmo.1	|- BOUNDED ph

Assertion

Ref	Expression
bdrmo	|- BOUNDED E* x e. y ph

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bdrmo

Step	Hyp	Ref	Expression
1		bdrmo.1	. . . 4 |- BOUNDED ph
2	1	ax-bdex 13006	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 13042	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 13001	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2644	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 13012	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2415   E!wreu 2416   E*wrmo 2417  BOUNDED wbd 12999

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-bd0 13000  ax-bdim 13001  ax-bdan 13002  ax-bdal 13005  ax-bdex 13006  ax-bdeq 13007

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-cleq 2130  df-clel 2133  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422

This theorem is referenced by: (None)