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Theorem bj-snmoore 34679
 Description: A singleton is a Moore collection. See bj-snmooreb 34680 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 4823 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
2 snidg 4562 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2eqeltrd 2890 . 2 (𝐴𝑉 {𝐴} ∈ {𝐴})
4 df-ne 2988 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
5 sssn 4722 . . . . . 6 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6 biorf 934 . . . . . . 7 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
76biimpar 481 . . . . . 6 ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
84, 5, 7syl2anb 600 . . . . 5 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
9 inteq 4845 . . . . . . 7 (𝑥 = {𝐴} → 𝑥 = {𝐴})
10 intsng 4877 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
11 eqtr 2818 . . . . . . . 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 5208 . . . . . . . 8 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
15 elsng 4542 . . . . . . . 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 34676 1 (𝐴𝑉 → {𝐴} ∈ Moore)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3442   ⊆ wss 3883  ∅c0 4246  {csn 4528  ∪ cuni 4804  ∩ cint 4842  Moorecmoore 34669 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-uni 4805  df-int 4843  df-bj-moore 34670 This theorem is referenced by:  bj-snmooreb  34680  bj-prmoore  34681
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