Theorem bdrmo 16219

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 16182	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 16218	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 16177	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2752	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 16188	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2509   E!wreu 2510   E*wrmo 2511  BOUNDED wbd 16175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16176  ax-bdim 16177  ax-bdan 16178  ax-bdal 16181  ax-bdex 16182  ax-bdeq 16183

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-cleq 2222  df-clel 2225  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516

This theorem is referenced by: (None)