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Theorem bj-prmoore 34445
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 34443). 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 pm3.22 463 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴𝑉) → (𝐴𝑉𝐵 ∈ V))
21adantrr 716 . . . . . 6 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → (𝐴𝑉𝐵 ∈ V))
3 uniprg 4843 . . . . . 6 ((𝐴𝑉𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
42, 3syl 17 . . . . 5 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} = (𝐴𝐵))
5 simprr 772 . . . . . 6 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → 𝐴𝐵)
6 ssequn1 4142 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
75, 6sylib 221 . . . . 5 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → (𝐴𝐵) = 𝐵)
84, 7eqtrd 2859 . . . 4 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} = 𝐵)
9 prid2g 4682 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ {𝐴, 𝐵})
109adantr 484 . . . 4 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → 𝐵 ∈ {𝐴, 𝐵})
118, 10eqeltrd 2916 . . 3 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} ∈ {𝐴, 𝐵})
12 biid 264 . . . . 5 ((𝐴𝑉𝐴𝐵) ↔ (𝐴𝑉𝐴𝐵))
1312bianass 641 . . . 4 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) ↔ ((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵))
14 inteq 4866 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 = {𝐴})
15 intsng 4898 . . . . . . . . . . 11 (𝐴𝑉 {𝐴} = 𝐴)
1615adantl 485 . . . . . . . . . 10 ((𝐵 ∈ V ∧ 𝐴𝑉) → {𝐴} = 𝐴)
1714, 16sylan9eqr 2881 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐴}) → 𝑥 = 𝐴)
18 prid1g 4681 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
1918adantl 485 . . . . . . . . . 10 ((𝐵 ∈ V ∧ 𝐴𝑉) → 𝐴 ∈ {𝐴, 𝐵})
2019adantr 484 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐴}) → 𝐴 ∈ {𝐴, 𝐵})
2117, 20eqeltrd 2916 . . . . . . . 8 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐴}) → 𝑥 ∈ {𝐴, 𝐵})
2221ex 416 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴𝑉) → (𝑥 = {𝐴} → 𝑥 ∈ {𝐴, 𝐵}))
2322adantr 484 . . . . . 6 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝑥 = {𝐴} → 𝑥 ∈ {𝐴, 𝐵}))
24 inteq 4866 . . . . . . . . . . 11 (𝑥 = {𝐵} → 𝑥 = {𝐵})
25 intsng 4898 . . . . . . . . . . . 12 (𝐵 ∈ V → {𝐵} = 𝐵)
2625adantr 484 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝐴𝑉) → {𝐵} = 𝐵)
2724, 26sylan9eqr 2881 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐵}) → 𝑥 = 𝐵)
289ad2antrr 725 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐵}) → 𝐵 ∈ {𝐴, 𝐵})
2927, 28eqeltrd 2916 . . . . . . . . 9 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝑥 = {𝐵}) → 𝑥 ∈ {𝐴, 𝐵})
3029ex 416 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝐴𝑉) → (𝑥 = {𝐵} → 𝑥 ∈ {𝐴, 𝐵}))
3130adantr 484 . . . . . . 7 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝑥 = {𝐵} → 𝑥 ∈ {𝐴, 𝐵}))
32 inteq 4866 . . . . . . . . . . 11 (𝑥 = {𝐴, 𝐵} → 𝑥 = {𝐴, 𝐵})
3332adantl 485 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝑥 = {𝐴, 𝐵})
341ad2antrr 725 . . . . . . . . . . 11 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → (𝐴𝑉𝐵 ∈ V))
35 intprg 4897 . . . . . . . . . . 11 ((𝐴𝑉𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
3634, 35syl 17 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → {𝐴, 𝐵} = (𝐴𝐵))
37 df-ss 3936 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
3837biimpi 219 . . . . . . . . . . . 12 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
3938adantl 485 . . . . . . . . . . 11 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝐴𝐵) = 𝐴)
4039adantr 484 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → (𝐴𝐵) = 𝐴)
4133, 36, 403eqtrd 2863 . . . . . . . . 9 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝑥 = 𝐴)
4218ad3antlr 730 . . . . . . . . 9 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝐴 ∈ {𝐴, 𝐵})
4341, 42eqeltrd 2916 . . . . . . . 8 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ 𝑥 = {𝐴, 𝐵}) → 𝑥 ∈ {𝐴, 𝐵})
4443ex 416 . . . . . . 7 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → (𝑥 = {𝐴, 𝐵} → 𝑥 ∈ {𝐴, 𝐵}))
4531, 44jaod 856 . . . . . 6 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → ((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) → 𝑥 ∈ {𝐴, 𝐵}))
4623, 45jaod 856 . . . . 5 (((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) → ((𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) → 𝑥 ∈ {𝐴, 𝐵}))
47 sspr 4751 . . . . . 6 (𝑥 ⊆ {𝐴, 𝐵} ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
48 andir 1006 . . . . . . 7 ((((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∧ 𝑥 ≠ ∅) ↔ (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) ∨ ((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) ∧ 𝑥 ≠ ∅)))
49 andir 1006 . . . . . . . . 9 (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) ↔ ((𝑥 = ∅ ∧ 𝑥 ≠ ∅) ∨ (𝑥 = {𝐴} ∧ 𝑥 ≠ ∅)))
50 eqneqall 3025 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 ≠ ∅ → ⊥))
5150imp 410 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑥 ≠ ∅) → ⊥)
52 simpl 486 . . . . . . . . . . 11 ((𝑥 = {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
5351, 52orim12i 906 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑥 ≠ ∅) ∨ (𝑥 = {𝐴} ∧ 𝑥 ≠ ∅)) → (⊥ ∨ 𝑥 = {𝐴}))
54 falim 1555 . . . . . . . . . . 11 (⊥ → 𝑥 = {𝐴})
5554bj-jaoi1 33931 . . . . . . . . . 10 ((⊥ ∨ 𝑥 = {𝐴}) → 𝑥 = {𝐴})
5653, 55syl 17 . . . . . . . . 9 (((𝑥 = ∅ ∧ 𝑥 ≠ ∅) ∨ (𝑥 = {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
5749, 56sylbi 220 . . . . . . . 8 (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
58 simpl 486 . . . . . . . 8 (((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) ∧ 𝑥 ≠ ∅) → (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))
5957, 58orim12i 906 . . . . . . 7 ((((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∧ 𝑥 ≠ ∅) ∨ ((𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}) ∧ 𝑥 ≠ ∅)) → (𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
6048, 59sylbi 220 . . . . . 6 ((((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∧ 𝑥 ≠ ∅) → (𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
6147, 60sylanb 584 . . . . 5 ((𝑥 ⊆ {𝐴, 𝐵} ∧ 𝑥 ≠ ∅) → (𝑥 = {𝐴} ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
6246, 61impel 509 . . . 4 ((((𝐵 ∈ V ∧ 𝐴𝑉) ∧ 𝐴𝐵) ∧ (𝑥 ⊆ {𝐴, 𝐵} ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ {𝐴, 𝐵})
6313, 62sylanb 584 . . 3 (((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) ∧ (𝑥 ⊆ {𝐴, 𝐵} ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ {𝐴, 𝐵})
6411, 63bj-ismooredr2 34440 . 2 ((𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} ∈ Moore)
65 pm3.22 463 . . . . 5 ((¬ 𝐵 ∈ V ∧ 𝐴𝑉) → (𝐴𝑉 ∧ ¬ 𝐵 ∈ V))
6665adantrr 716 . . . 4 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → (𝐴𝑉 ∧ ¬ 𝐵 ∈ V))
67 prprc2 4687 . . . . . 6 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
6867adantl 485 . . . . 5 ((𝐴𝑉 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = {𝐴})
6968eqcomd 2830 . . . 4 ((𝐴𝑉 ∧ ¬ 𝐵 ∈ V) → {𝐴} = {𝐴, 𝐵})
7066, 69syl 17 . . 3 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴} = {𝐴, 𝐵})
71 bj-snmoore 34443 . . . 4 (𝐴𝑉 → {𝐴} ∈ Moore)
7271ad2antrl 727 . . 3 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴} ∈ Moore)
7370, 72eqeltrrd 2917 . 2 ((¬ 𝐵 ∈ V ∧ (𝐴𝑉𝐴𝐵)) → {𝐴, 𝐵} ∈ Moore)
7464, 73pm2.61ian 811 1 ((𝐴𝑉𝐴𝐵) → {𝐴, 𝐵} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wfal 1550  wcel 2115  wne 3014  Vcvv 3480  cun 3917  cin 3918  wss 3919  c0 4276  {csn 4550  {cpr 4552   cuni 4825   cint 4863  Moorecmoore 34433
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553  df-uni 4826  df-int 4864  df-bj-moore 34434
This theorem is referenced by: (None)
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