Theorem bdrmo 14148

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 14111	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 14147	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 14106	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2690	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 14117	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2454   E!wreu 2455   E*wrmo 2456  BOUNDED wbd 14104

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-bd0 14105  ax-bdim 14106  ax-bdan 14107  ax-bdal 14110  ax-bdex 14111  ax-bdeq 14112

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-cleq 2168  df-clel 2171  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461

This theorem is referenced by: (None)