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Theorem bj-prmoore 35586
Description: A pair formed of two nested sets is a Moore collection. (Note that in the statement, if 𝐵 is a proper class, we are in the case of bj-snmoore 35584). A direct consequence is {∅, 𝐴} ∈ Moore.

More generally, any nonempty well-ordered chain of sets that is a set is a Moore collection.

We also have the biconditional ((𝐴𝐵) ∈ 𝑉 ({𝐴, 𝐵} ∈ Moore ↔ (𝐴𝐵𝐵𝐴))). (Contributed by BJ, 11-Apr-2024.)

Assertion
Ref Expression
bj-prmoore ((𝐴𝑉𝐴𝐵) → {𝐴, 𝐵} ∈ Moore)

Proof of Theorem bj-prmoore
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm3.22 460 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴𝑉) → (𝐴𝑉𝐵 ∈ V))
21adantrr 715 . . . . . 6 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → (𝐴𝑉𝐵 ∈ V))
3 uniprg 4882 . . . . . 6 ((𝐴𝑉𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
42, 3syl 17 . . . . 5 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} = (𝐴𝐵))
5 simprr 771 . . . . . 6 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → 𝐴𝐵)
6 ssequn1 4140 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
75, 6sylib 217 . . . . 5 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → (𝐴𝐵) = 𝐵)
84, 7eqtrd 2776 . . . 4 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} = 𝐵)
9 prid2g 4722 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ {𝐴, 𝐵})
109adantr 481 . . . 4 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → 𝐵 ∈ {𝐴, 𝐵})
118, 10eqeltrd 2838 . . 3 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} ∈ {𝐴, 𝐵})
12 biid 260 . . . . 5 ((𝐴𝑉𝐴𝐵) ↔ (𝐴𝑉𝐴𝐵))
1312bianass 640 . . . 4 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) ↔ ((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵))
14 inteq 4910 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 = {𝐴})
15 intsng 4946 . . . . . . . . . . 11 (𝐴𝑉 {𝐴} = 𝐴)
1615adantl 482 . . . . . . . . . 10 ((𝐵 ∈ V ∧ 𝐴𝑉) → {𝐴} = 𝐴)
1714, 16sylan9eqr 2798 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐴}) → 𝑥 = 𝐴)
18 prid1g 4721 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
1918adantl 482 . . . . . . . . . 10 ((𝐵 ∈ V ∧ 𝐴𝑉) → 𝐴 ∈ {𝐴, 𝐵})
2019adantr 481 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐴}) → 𝐴 ∈ {𝐴, 𝐵})
2117, 20eqeltrd 2838 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐴}) → 𝑥 ∈ {𝐴, 𝐵})
2221ex 413 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴𝑉) → (𝑥 = {𝐴} → 𝑥 ∈ {𝐴, 𝐵}))
2322adantr 481 . . . . . 6 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝑥 = {𝐴} → 𝑥 ∈ {𝐴, 𝐵}))
24 inteq 4910 . . . . . . . . . . 11 (𝑥 = {𝐵} → 𝑥 = {𝐵})
25 intsng 4946 . . . . . . . . . . . 12 (𝐵 ∈ V → {𝐵} = 𝐵)
2625adantr 481 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝐴𝑉) → {𝐵} = 𝐵)
2724, 26sylan9eqr 2798 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐵}) → 𝑥 = 𝐵)
289ad2antrr 724 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐵}) → 𝐵 ∈ {𝐴, 𝐵})
2927, 28eqeltrd 2838 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐵}) → 𝑥 ∈ {𝐴, 𝐵})
3029ex 413 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝐴𝑉) → (𝑥 = {𝐵} → 𝑥 ∈ {𝐴, 𝐵}))
3130adantr 481 . . . . . . 7 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝑥 = {𝐵} → 𝑥 ∈ {𝐴, 𝐵}))
32 inteq 4910 . . . . . . . . . . 11 (𝑥 = {𝐴, 𝐵} → 𝑥 = {𝐴, 𝐵})
3332adantl 482 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝑥 = {𝐴, 𝐵})
341ad2antrr 724 . . . . . . . . . . 11 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → (𝐴𝑉𝐵 ∈ V))
35 intprg 4942 . . . . . . . . . . 11 ((𝐴𝑉𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
3634, 35syl 17 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → {𝐴, 𝐵} = (𝐴𝐵))
37 df-ss 3927 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
3837biimpi 215 . . . . . . . . . . . 12 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
3938adantl 482 . . . . . . . . . . 11 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝐴𝐵) = 𝐴)
4039adantr 481 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → (𝐴𝐵) = 𝐴)
4133, 36, 403eqtrd 2780 . . . . . . . . 9 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝑥 = 𝐴)
4218ad3antlr 729 . . . . . . . . 9 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝐴 ∈ {𝐴, 𝐵})
4341, 42eqeltrd 2838 . . . . . . . 8 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝑥 ∈ {𝐴, 𝐵})
4443ex 413 . . . . . . 7 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝑥 = {𝐴, 𝐵} → 𝑥 ∈ {𝐴, 𝐵}))
4531, 44jaod 857 . . . . . 6 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → ((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) → 𝑥 ∈ {𝐴, 𝐵}))
4623, 45jaod 857 . . . . 5 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → ((𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) → 𝑥 ∈ {𝐴, 𝐵}))
47 sspr 4793 . . . . . 6 (𝑥 ⊆ {𝐴, 𝐵} ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
48 andir 1007 . . . . . . 7 ((((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∧ 𝑥 ≠ ∅) ↔ (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) ∨ ((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) ∧ 𝑥 ≠ ∅)))
49 andir 1007 . . . . . . . . 9 (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) ↔ ((𝑥 = ∅ ∧ 𝑥 ≠ ∅) ∨ (𝑥 = {𝐴} ∧ 𝑥 ≠ ∅)))
50 eqneqall 2954 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 ≠ ∅ → ⊥))
5150imp 407 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑥 ≠ ∅) → ⊥)
52 simpl 483 . . . . . . . . . . 11 ((𝑥 = {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
5351, 52orim12i 907 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑥 ≠ ∅) ∨ (𝑥 = {𝐴} ∧ 𝑥 ≠ ∅)) → (⊥ ∨ 𝑥 = {𝐴}))
54 falim 1558 . . . . . . . . . . 11 (⊥ → 𝑥 = {𝐴})
5554bj-jaoi1 35035 . . . . . . . . . 10 ((⊥ ∨ 𝑥 = {𝐴}) → 𝑥 = {𝐴})
5653, 55syl 17 . . . . . . . . 9 (((𝑥 = ∅ ∧ 𝑥 ≠ ∅) ∨ (𝑥 = {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
5749, 56sylbi 216 . . . . . . . 8 (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
58 simpl 483 . . . . . . . 8 (((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) ∧ 𝑥 ≠ ∅) → (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))
5957, 58orim12i 907 . . . . . . 7 ((((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) ∨ ((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) ∧ 𝑥 ≠ ∅)) → (𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
6048, 59sylbi 216 . . . . . 6 ((((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∧ 𝑥 ≠ ∅) → (𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
6147, 60sylanb 581 . . . . 5 ((𝑥 ⊆ {𝐴, 𝐵} ∧ 𝑥 ≠ ∅) → (𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
6246, 61impel 506 . . . 4 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ (𝑥 ⊆ {𝐴, 𝐵} ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ {𝐴, 𝐵})
6313, 62sylanb 581 . . 3 (((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) ∧ (𝑥 ⊆ {𝐴, 𝐵} ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ {𝐴, 𝐵})
6411, 63bj-ismooredr2 35581 . 2 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} ∈ Moore)
65 pm3.22 460 . . . . 5 ((¬ 𝐵 ∈ V ∧ 𝐴𝑉) → (𝐴𝑉 ∧ ¬ 𝐵 ∈ V))
6665adantrr 715 . . . 4 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → (𝐴𝑉 ∧ ¬ 𝐵 ∈ V))
67 prprc2 4727 . . . . . 6 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
6867adantl 482 . . . . 5 ((𝐴𝑉 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = {𝐴})
6968eqcomd 2742 . . . 4 ((𝐴𝑉 ∧ ¬ 𝐵 ∈ V) → {𝐴} = {𝐴, 𝐵})
7066, 69syl 17 . . 3 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴} = {𝐴, 𝐵})
71 bj-snmoore 35584 . . . 4 (𝐴𝑉 → {𝐴} ∈ Moore)
7271ad2antrl 726 . . 3 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴} ∈ Moore)
7370, 72eqeltrrd 2839 . 2 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} ∈ Moore)
7464, 73pm2.61ian 810 1 ((𝐴𝑉𝐴𝐵) → {𝐴, 𝐵} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wfal 1553  wcel 2106  wne 2943  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  {cpr 4588   cuni 4865   cint 4907  Moorecmoore 35574
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320
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 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-pw 4562  df-sn 4587  df-pr 4589  df-uni 4866  df-int 4908  df-bj-moore 35575
This theorem is referenced by: (None)
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