Theorem bdrmo 13225

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 13188	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3	1	bdreu 13224	. . 3 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
4	2, 3	ax-bdim 13183	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑)
5		rmo5 2649	. 2 ⊢ (∃*𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑))
6	4, 5	bd0r 13194	1 ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4  ∃wrex 2418  ∃!wreu 2419  ∃*wrmo 2420  BOUNDED wbd 13181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bd0 13182  ax-bdim 13183  ax-bdan 13184  ax-bdal 13187  ax-bdex 13188  ax-bdeq 13189

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-cleq 2133  df-clel 2136  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425

This theorem is referenced by: (None)