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Theorem bj-snmoore 33400
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 5037 . . . 4 {𝐴} ∈ V
21a1i 11 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
3 unisng 4591 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
4 snidg 4346 . . . 4 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
53, 4eqeltrd 2850 . . 3 (𝐴 ∈ V → {𝐴} ∈ {𝐴})
6 df-ne 2944 . . . . . . . 8 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
7 sssn 4493 . . . . . . . 8 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
8 biorf 922 . . . . . . . . 9 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
98biimpar 463 . . . . . . . 8 ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
106, 7, 9syl2anb 585 . . . . . . 7 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
11 inteq 4615 . . . . . . . . 9 (𝑥 = {𝐴} → 𝑥 = {𝐴})
12 intsng 4647 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} = 𝐴)
13 eqtr 2790 . . . . . . . . . 10 (( 𝑥 = {𝐴} ∧ {𝐴} = 𝐴) → 𝑥 = 𝐴)
1413ex 397 . . . . . . . . 9 ( 𝑥 = {𝐴} → ( {𝐴} = 𝐴 𝑥 = 𝐴))
1511, 12, 14syl2im 40 . . . . . . . 8 (𝑥 = {𝐴} → (𝐴 ∈ V → 𝑥 = 𝐴))
16 intex 4952 . . . . . . . . . 10 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
17 elsng 4331 . . . . . . . . . 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 498 . . . . . . 7 ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ V → 𝑥 ∈ {𝐴}))
2110, 20syldan 579 . . . . . 6 ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ V → 𝑥 ∈ {𝐴}))
2221ex 397 . . . . 5 (𝑥 ≠ ∅ → (𝑥 ⊆ {𝐴} → (𝐴 ∈ V → 𝑥 ∈ {𝐴})))
2322com13 88 . . . 4 (𝐴 ∈ V → (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 ∈ {𝐴})))
2423imp31 404 . . 3 (((𝐴 ∈ V ∧ 𝑥 ⊆ {𝐴}) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ {𝐴})
252, 5, 24bj-ismooredr2 33397 . 2 (𝐴 ∈ V → {𝐴} ∈ Moore)
26 snprc 4390 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
2726biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
28 bj-0nmoore 33399 . . . . 5 ¬ ∅ ∈ Moore
2928a1i 11 . . . 4 𝐴 ∈ V → ¬ ∅ ∈ Moore)
3027, 29eqneltrd 2869 . . 3 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
3130con4i 114 . 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 836   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  wss 3723  c0 4063  {csn 4317   cuni 4575   cint 4612  Moorecmoore 33389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4300  df-sn 4318  df-pr 4320  df-uni 4576  df-int 4613  df-bj-moore 33390
This theorem is referenced by: (None)
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