Theorem bdrmo 16451

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 16414	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3	1	bdreu 16450	. . 3 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
4	2, 3	ax-bdim 16409	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑)
5		rmo5 2754	. 2 ⊢ (∃*𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑))
6	4, 5	bd0r 16420	1 ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4  ∃wrex 2511  ∃!wreu 2512  ∃*wrmo 2513  BOUNDED wbd 16407

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bd0 16408  ax-bdim 16409  ax-bdan 16410  ax-bdal 16413  ax-bdex 16414  ax-bdeq 16415

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-cleq 2224  df-clel 2227  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518

This theorem is referenced by: (None)