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Theorem bj-snmoore 35796
Description: A singleton is a Moore collection. See bj-snmooreb 35797 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 4922 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
2 snidg 4656 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2eqeltrd 2832 . 2 (𝐴𝑉 {𝐴} ∈ {𝐴})
4 df-ne 2940 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
5 sssn 4822 . . . . . 6 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6 biorf 935 . . . . . . 7 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
76biimpar 478 . . . . . 6 ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
84, 5, 7syl2anb 598 . . . . 5 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
9 inteq 4946 . . . . . . 7 (𝑥 = {𝐴} → 𝑥 = {𝐴})
10 intsng 4982 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
11 eqtr 2754 . . . . . . . 8 (( 𝑥 = {𝐴} ∧ {𝐴} = 𝐴) → 𝑥 = 𝐴)
1211ex 413 . . . . . . 7 ( 𝑥 = {𝐴} → ( {𝐴} = 𝐴 𝑥 = 𝐴))
139, 10, 12syl2im 40 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑉 𝑥 = 𝐴))
14 intex 5330 . . . . . . . 8 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
15 elsng 4636 . . . . . . . 8 ( 𝑥 ∈ V → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1614, 15sylbi 216 . . . . . . 7 (𝑥 ≠ ∅ → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1716biimprd 247 . . . . . 6 (𝑥 ≠ ∅ → ( 𝑥 = 𝐴 𝑥 ∈ {𝐴}))
1813, 17sylan9r 509 . . . . 5 ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴𝑉 𝑥 ∈ {𝐴}))
198, 18syldan 591 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴𝑉 𝑥 ∈ {𝐴}))
2019ancoms 459 . . 3 ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → (𝐴𝑉 𝑥 ∈ {𝐴}))
2120impcom 408 . 2 ((𝐴𝑉 ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ {𝐴})
223, 21bj-ismooredr2 35793 1 (𝐴𝑉 → {𝐴} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2939  Vcvv 3473  wss 3944  c0 4318  {csn 4622   cuni 4901   cint 4943  Moorecmoore 35786
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-pw 4598  df-sn 4623  df-pr 4625  df-uni 4902  df-int 4944  df-bj-moore 35787
This theorem is referenced by:  bj-snmooreb  35797  bj-prmoore  35798
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