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Theorem bj-snmoore 35019
Description: A singleton is a Moore collection. See bj-snmooreb 35020 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 4840 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
2 snidg 4575 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2eqeltrd 2838 . 2 (𝐴𝑉 {𝐴} ∈ {𝐴})
4 df-ne 2941 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
5 sssn 4739 . . . . . 6 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6 biorf 937 . . . . . . 7 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
76biimpar 481 . . . . . 6 ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
84, 5, 7syl2anb 601 . . . . 5 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
9 inteq 4862 . . . . . . 7 (𝑥 = {𝐴} → 𝑥 = {𝐴})
10 intsng 4896 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
11 eqtr 2760 . . . . . . . 8 (( 𝑥 = {𝐴} ∧ {𝐴} = 𝐴) → 𝑥 = 𝐴)
1211ex 416 . . . . . . 7 ( 𝑥 = {𝐴} → ( {𝐴} = 𝐴 𝑥 = 𝐴))
139, 10, 12syl2im 40 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑉 𝑥 = 𝐴))
14 intex 5230 . . . . . . . 8 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
15 elsng 4555 . . . . . . . 8 ( 𝑥 ∈ V → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1614, 15sylbi 220 . . . . . . 7 (𝑥 ≠ ∅ → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1716biimprd 251 . . . . . 6 (𝑥 ≠ ∅ → ( 𝑥 = 𝐴 𝑥 ∈ {𝐴}))
1813, 17sylan9r 512 . . . . 5 ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴𝑉 𝑥 ∈ {𝐴}))
198, 18syldan 594 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴𝑉 𝑥 ∈ {𝐴}))
2019ancoms 462 . . 3 ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → (𝐴𝑉 𝑥 ∈ {𝐴}))
2120impcom 411 . 2 ((𝐴𝑉 ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ {𝐴})
223, 21bj-ismooredr2 35016 1 (𝐴𝑉 → {𝐴} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  wne 2940  Vcvv 3408  wss 3866  c0 4237  {csn 4541   cuni 4819   cint 4859  Moorecmoore 35009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-pw 4515  df-sn 4542  df-pr 4544  df-uni 4820  df-int 4860  df-bj-moore 35010
This theorem is referenced by:  bj-snmooreb  35020  bj-prmoore  35021
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