Theorem rmosn 4724

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 3811	. . . . 5 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
3		sbceq1a 3802	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
4	2, 3	rexsngf 4677	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	2, 3	reusngf 4679	. . . 4 ⊢ (𝐴 ∈ V → (∃!𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	1, 4, 5	3imtr4d 294	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
7		rmo5 3398	. . 3 ⊢ (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
8	6, 7	sylibr 234	. 2 ⊢ (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
9		rmo0 4368	. . 3 ⊢ ∃*𝑥 ∈ ∅ 𝜑
10		snprc 4722	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11		rmoeq1 3414	. . . 4 ⊢ ({𝐴} = ∅ → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
12	10, 11	sylbi 217	. . 3 ⊢ (¬ 𝐴 ∈ V → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
13	9, 12	mpbiri 258	. 2 ⊢ (¬ 𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
14	8, 13	pm2.61i 182	1 ⊢ ∃*𝑥 ∈ {𝐴}𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  ∃!wreu 3376  ∃*wrmo 3377  Vcvv 3478  [wsbc 3791  ∅c0 4339  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-v 3480  df-sbc 3792  df-dif 3966  df-nul 4340  df-sn 4632

This theorem is referenced by:  mosn  48661