Theorem bdrmo 14961

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 14924	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 14960	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 14919	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2703	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 14930	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2466   E!wreu 2467   E*wrmo 2468  BOUNDED wbd 14917

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bd0 14918  ax-bdim 14919  ax-bdan 14920  ax-bdal 14923  ax-bdex 14924  ax-bdeq 14925

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-cleq 2180  df-clel 2183  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473

This theorem is referenced by: (None)