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Theorem bj-snmoore 37096
Description: A singleton is a Moore collection. See bj-snmooreb 37097 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
Assertion
Ref Expression
bj-snmoore (𝐴𝑉 → {𝐴} ∈ Moore)

Proof of Theorem bj-snmoore
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 unisng 4930 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
2 snidg 4665 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2eqeltrd 2839 . 2 (𝐴𝑉 {𝐴} ∈ {𝐴})
4 df-ne 2939 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
5 sssn 4831 . . . . . 6 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6 biorf 936 . . . . . . 7 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
76biimpar 477 . . . . . 6 ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
84, 5, 7syl2anb 598 . . . . 5 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
9 inteq 4954 . . . . . . 7 (𝑥 = {𝐴} → 𝑥 = {𝐴})
10 intsng 4988 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
11 eqtr 2758 . . . . . . . 8 (( 𝑥 = {𝐴} ∧ {𝐴} = 𝐴) → 𝑥 = 𝐴)
1211ex 412 . . . . . . 7 ( 𝑥 = {𝐴} → ( {𝐴} = 𝐴 𝑥 = 𝐴))
139, 10, 12syl2im 40 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑉 𝑥 = 𝐴))
14 intex 5350 . . . . . . . 8 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
15 elsng 4645 . . . . . . . 8 ( 𝑥 ∈ V → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1614, 15sylbi 217 . . . . . . 7 (𝑥 ≠ ∅ → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1716biimprd 248 . . . . . 6 (𝑥 ≠ ∅ → ( 𝑥 = 𝐴 𝑥 ∈ {𝐴}))
1813, 17sylan9r 508 . . . . 5 ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴𝑉 𝑥 ∈ {𝐴}))
198, 18syldan 591 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴𝑉 𝑥 ∈ {𝐴}))
2019ancoms 458 . . 3 ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → (𝐴𝑉 𝑥 ∈ {𝐴}))
2120impcom 407 . 2 ((𝐴𝑉 ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ {𝐴})
223, 21bj-ismooredr2 37093 1 (𝐴𝑉 → {𝐴} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  wss 3963  c0 4339  {csn 4631   cuni 4912   cint 4951  Moorecmoore 37086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-int 4952  df-bj-moore 37087
This theorem is referenced by:  bj-snmooreb  37097  bj-prmoore  37098
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