Theorem rmosn 4675

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 24	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
2		nfsbc1v 3762	. . . . 5 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
3		sbceq1a 3753	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
4	2, 3	rexsngf 4628	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	2, 3	reusngf 4630	. . . 4 ⊢ (𝐴 ∈ V → (∃!𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	1, 4, 5	3imtr4d 296	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
7		rmo5 3384	. . 3 ⊢ (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
8	6, 7	sylibr 236	. 2 ⊢ (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
9		rmo0 4312	. . 3 ⊢ ∃*𝑥 ∈ ∅ 𝜑
10		snprc 4673	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11		rmoeq1 3397	. . . 4 ⊢ ({𝐴} = ∅ → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
12	10, 11	sylbi 219	. . 3 ⊢ (¬ 𝐴 ∈ V → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
13	9, 12	mpbiri 260	. 2 ⊢ (¬ 𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
14	8, 13	pm2.61i 183	1 ⊢ ∃*𝑥 ∈ {𝐴}𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ∃!wreu 3364  ∃*wrmo 3365  Vcvv 3453  [wsbc 3742  ∅c0 4283  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-v 3455  df-sbc 3743  df-dif 3905  df-nul 4284  df-sn 4580

This theorem is referenced by:  mosn  49395