Theorem bdrmo 16572

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 16535	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 16571	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 16530	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2755	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 16541	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2512   E!wreu 2513   E*wrmo 2514  BOUNDED wbd 16528

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-bd0 16529  ax-bdim 16530  ax-bdan 16531  ax-bdal 16534  ax-bdex 16535  ax-bdeq 16536

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-cleq 2224  df-clel 2227  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519

This theorem is referenced by: (None)