Theorem bdrmo 15348

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 15311	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 15347	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 15306	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2714	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 15317	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2473   E!wreu 2474   E*wrmo 2475  BOUNDED wbd 15304

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 2175  ax-bd0 15305  ax-bdim 15306  ax-bdan 15307  ax-bdal 15310  ax-bdex 15311  ax-bdeq 15312

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-cleq 2186  df-clel 2189  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480

This theorem is referenced by: (None)