Theorem mosn 49922

Description: "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
mosn	⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mosn

Step	Hyp	Ref	Expression
1		rmosn 4680	. . 3 ⊢ ∃*𝑥 ∈ {𝐵}⊤
2		rmotru 49912	. . 3 ⊢ (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤)
3	1, 2	mpbir 234	. 2 ⊢ ∃*𝑥 𝑥 ∈ {𝐵}
4		eleq2 2850	. . 3 ⊢ (𝐴 = {𝐵} → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵}))
5	4	mobidv 2575	. 2 ⊢ (𝐴 = {𝐵} → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵}))
6	3, 5	mpbiri 261	1 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  ∃*wmo 2563  ∃*wrmo 3365  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-v 3453  df-sbc 3740  df-dif 3902  df-nul 4280  df-sn 4585

This theorem is used by:  mo0  49923  mosssn  49924  mo0sn  49925  f1omo  50000  oppcmndclem  50124  indcthing  50567  discthing  50568  termcbasmo  50590  setcsnterm  50597  idfudiag1  50632