Theorem rmosn 4687

Description: A restricted at-most-one quantifier over a singleton is always true. (Contributed by AV, 3-Apr-2023.)

Assertion

Ref	Expression
rmosn	⊢ ∃*𝑥 ∈ {𝐴}𝜑

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmosn

Step	Hyp	Ref	Expression
1		idd 25	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
2		nfsbc1v 3773	. . . . 5 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
3		sbceq1a 3764	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
4	2, 3	rexsngf 4640	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	2, 3	reusngf 4642	. . . 4 ⊢ (𝐴 ∈ V → (∃!𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	1, 4, 5	3imtr4d 297	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
7		rmo5 3394	. . 3 ⊢ (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
8	6, 7	sylibr 237	. 2 ⊢ (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
9		rmo0 4324	. . 3 ⊢ ∃*𝑥 ∈ ∅ 𝜑
10		snprc 4685	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11		rmoeq1 3407	. . . 4 ⊢ ({𝐴} = ∅ → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
12	10, 11	sylbi 220	. . 3 ⊢ (¬ 𝐴 ∈ V → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
13	9, 12	mpbiri 261	. 2 ⊢ (¬ 𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
14	8, 13	pm2.61i 184	1 ⊢ ∃*𝑥 ∈ {𝐴}𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  ∃!wreu 3374  ∃*wrmo 3375  Vcvv 3463  [wsbc 3753  ∅c0 4294  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-v 3465  df-sbc 3754  df-dif 3916  df-nul 4295  df-sn 4592

This theorem is referenced by:  mosn  49469