Theorem bdrmo 14991

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 14954	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3	1	bdreu 14990	. . 3 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
4	2, 3	ax-bdim 14949	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑)
5		rmo5 2705	. 2 ⊢ (∃*𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑))
6	4, 5	bd0r 14960	1 ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4  ∃wrex 2468  ∃!wreu 2469  ∃*wrmo 2470  BOUNDED wbd 14947

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 2170  ax-bd0 14948  ax-bdim 14949  ax-bdan 14950  ax-bdal 14953  ax-bdex 14954  ax-bdeq 14955

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-cleq 2181  df-clel 2184  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475

This theorem is referenced by: (None)