Theorem mosn 49303

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 4651	. . 3 ⊢ ∃*𝑥 ∈ {𝐵}⊤
2		rmotru 49293	. . 3 ⊢ (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤)
3	1, 2	mpbir 232	. 2 ⊢ ∃*𝑥 𝑥 ∈ {𝐵}
4		eleq2 2828	. . 3 ⊢ (𝐴 = {𝐵} → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵}))
5	4	mobidv 2553	. 2 ⊢ (𝐴 = {𝐵} → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵}))
6	3, 5	mpbiri 259	1 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  ∃*wmo 2541  ∃*wrmo 3343  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-v 3433  df-sbc 3724  df-dif 3886  df-nul 4262  df-sn 4556

This theorem is referenced by:  mo0  49304  mosssn  49305  mo0sn  49306  f1omo  49383  oppcmndclem  49507  indcthing  49950  discthing  49951  termcbasmo  49973  setcsnterm  49980  idfudiag1  50015