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Theorem rmosn 4615
 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
StepHypRef Expression
1 idd 24 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 nfsbc1v 3718 . . . . 5 𝑥[𝐴 / 𝑥]𝜑
3 sbceq1a 3709 . . . . 5 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
42, 3rexsngf 4570 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
52, 3reusngf 4572 . . . 4 (𝐴 ∈ V → (∃!𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
61, 4, 53imtr4d 297 . . 3 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
7 rmo5 3344 . . 3 (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
86, 7sylibr 237 . 2 (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
9 rmo0 4260 . . 3 ∃*𝑥 ∈ ∅ 𝜑
10 snprc 4613 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
11 rmoeq1 3326 . . . 4 ({𝐴} = ∅ → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
1210, 11sylbi 220 . . 3 𝐴 ∈ V → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
139, 12mpbiri 261 . 2 𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
148, 13pm2.61i 185 1 ∃*𝑥 ∈ {𝐴}𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  ∃!wreu 3072  ∃*wrmo 3073  Vcvv 3409  [wsbc 3698  ∅c0 4227  {csn 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-v 3411  df-sbc 3699  df-dif 3863  df-nul 4228  df-sn 4526 This theorem is referenced by: (None)
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