Theorem bdrmo 15094

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 15057	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 15093	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 15052	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2706	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 15063	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2469   E!wreu 2470   E*wrmo 2471  BOUNDED wbd 15050

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15051  ax-bdim 15052  ax-bdan 15053  ax-bdal 15056  ax-bdex 15057  ax-bdeq 15058

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-cleq 2182  df-clel 2185  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476

This theorem is referenced by: (None)