Theorem bdrmo 10647

Description: Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdrmo.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdrmo	⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bdrmo

Step	Hyp	Ref	Expression
1		bdrmo.1	. . . 4 ⊢ BOUNDED 𝜑
2	1	ax-bdex 10610	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3	1	bdreu 10646	. . 3 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
4	2, 3	ax-bdim 10605	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑)
5		rmo5 2569	. 2 ⊢ (∃*𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑))
6	4, 5	bd0r 10616	1 ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4  ∃wrex 2349  ∃!wreu 2350  ∃*wrmo 2351  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdim 10605  ax-bdan 10606  ax-bdal 10609  ax-bdex 10610  ax-bdeq 10611

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-cleq 2074  df-clel 2077  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356

This theorem is referenced by: (None)