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Theorem bj-snmoore 33193
Description: A singleton is a Moore collection. (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-snmoore (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

Proof of Theorem bj-snmoore
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 snex 4938 . . . 4 {𝐴} ∈ V
21a1i 11 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
3 unisng 4484 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
4 snidg 4239 . . . 4 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
53, 4eqeltrd 2730 . . 3 (𝐴 ∈ V → {𝐴} ∈ {𝐴})
6 df-ne 2824 . . . . . . . 8 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
7 sssn 4390 . . . . . . . 8 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
8 biorf 419 . . . . . . . . 9 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
98biimpar 501 . . . . . . . 8 ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
106, 7, 9syl2anb 495 . . . . . . 7 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
11 inteq 4510 . . . . . . . . 9 (𝑥 = {𝐴} → 𝑥 = {𝐴})
12 intsng 4544 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} = 𝐴)
13 eqtr 2670 . . . . . . . . . 10 (( 𝑥 = {𝐴} ∧ {𝐴} = 𝐴) → 𝑥 = 𝐴)
1413ex 449 . . . . . . . . 9 ( 𝑥 = {𝐴} → ( {𝐴} = 𝐴 𝑥 = 𝐴))
1511, 12, 14syl2im 40 . . . . . . . 8 (𝑥 = {𝐴} → (𝐴 ∈ V → 𝑥 = 𝐴))
16 intex 4850 . . . . . . . . . 10 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
17 elsng 4224 . . . . . . . . . 10 ( 𝑥 ∈ V → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1816, 17sylbi 207 . . . . . . . . 9 (𝑥 ≠ ∅ → ( 𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
1918biimprd 238 . . . . . . . 8 (𝑥 ≠ ∅ → ( 𝑥 = 𝐴 𝑥 ∈ {𝐴}))
2015, 19sylan9r 691 . . . . . . 7 ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ V → 𝑥 ∈ {𝐴}))
2110, 20syldan 486 . . . . . 6 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ V → 𝑥 ∈ {𝐴}))
2221ex 449 . . . . 5 (𝑥 ≠ ∅ → (𝑥 ⊆ {𝐴} → (𝐴 ∈ V → 𝑥 ∈ {𝐴})))
2322com13 88 . . . 4 (𝐴 ∈ V → (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 ∈ {𝐴})))
2423imp31 447 . . 3 (((𝐴 ∈ V ∧ 𝑥 ⊆ {𝐴}) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ {𝐴})
252, 5, 24bj-ismooredr2 33190 . 2 (𝐴 ∈ V → {𝐴} ∈ Moore)
26 snprc 4285 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
2726biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
28 bj-0nmoore 33192 . . . . 5 ¬ ∅ ∈ Moore
2928a1i 11 . . . 4 𝐴 ∈ V → ¬ ∅ ∈ Moore)
3027, 29eqneltrd 2749 . . 3 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
3130con4i 113 . 2 ({𝐴} ∈ Moore𝐴 ∈ V)
3225, 31impbii 199 1 (𝐴 ∈ V ↔ {𝐴} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  wss 3607  c0 3948  {csn 4210   cuni 4468   cint 4507  Moorecmoore 33182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-int 4508  df-bj-moore 33183
This theorem is referenced by: (None)
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