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Theorem mosn 48737
Description: "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mosn (𝐴 = {𝐵} → ∃*𝑥 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mosn
StepHypRef Expression
1 rmosn 4718 . . 3 ∃*𝑥 ∈ {𝐵}⊤
2 rmotru 48728 . . 3 (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤)
31, 2mpbir 231 . 2 ∃*𝑥 𝑥 ∈ {𝐵}
4 eleq2 2829 . . 3 (𝐴 = {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
54mobidv 2548 . 2 (𝐴 = {𝐵} → (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵}))
63, 5mpbiri 258 1 (𝐴 = {𝐵} → ∃*𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wtru 1540  wcel 2107  ∃*wmo 2537  ∃*wrmo 3378  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-v 3481  df-sbc 3788  df-dif 3953  df-nul 4333  df-sn 4626
This theorem is referenced by:  mo0  48738  mosssn  48739  mo0sn  48740  oppcmndclem  48920  termcbasmo  49154  setcsnterm  49161  idfudiag1  49183
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