Theorem rmosn 4657

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 3794	. . . . 5 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
3		sbceq1a 3785	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
4	2, 3	rexsngf 4612	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	2, 3	reusngf 4614	. . . 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 3436	. . 3 ⊢ (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
8	6, 7	sylibr 236	. 2 ⊢ (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
9		rmo0 4321	. . 3 ⊢ ∃*𝑥 ∈ ∅ 𝜑
10		snprc 4655	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11		rmoeq1 3410	. . . 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 184	1 ⊢ ∃*𝑥 ∈ {𝐴}𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  ∃!wreu 3142  ∃*wrmo 3143  Vcvv 3496  [wsbc 3774  ∅c0 4293  {csn 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-v 3498  df-sbc 3775  df-dif 3941  df-nul 4294  df-sn 4570

This theorem is referenced by: (None)