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Theorem bj-inexeqex 34459
Description: Lemma for bj-opelid 34461 (but not specific to the identity relation): if the intersection of two classes is a set and the two classes are equal, then both are sets (all three classes are equal, so they all belong to 𝑉, but it is more convenient to have V in the consequent for theorems using it). (Contributed by BJ, 27-Dec-2023.)
Assertion
Ref Expression
bj-inexeqex (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem bj-inexeqex
StepHypRef Expression
1 eqimss 3998 . . . . 5 (𝐴 = 𝐵𝐴𝐵)
2 df-ss 3926 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
31, 2sylib 220 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
4 eleq1 2898 . . . . 5 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∈ 𝑉𝐴𝑉))
54biimpac 481 . . . 4 (((𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = 𝐴) → 𝐴𝑉)
63, 5sylan2 594 . . 3 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → 𝐴𝑉)
76elexd 3490 . 2 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
8 eqimss2 3999 . . . . 5 (𝐴 = 𝐵𝐵𝐴)
9 sseqin2 4166 . . . . 5 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
108, 9sylib 220 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐵)
11 eleq1 2898 . . . . 5 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) ∈ 𝑉𝐵𝑉))
1211biimpac 481 . . . 4 (((𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = 𝐵) → 𝐵𝑉)
1310, 12sylan2 594 . . 3 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → 𝐵𝑉)
1413elexd 3490 . 2 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → 𝐵 ∈ V)
157, 14jca 514 1 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  Vcvv 3470  cin 3908  wss 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-rab 3134  df-v 3472  df-in 3916  df-ss 3926
This theorem is referenced by:  bj-elsn0  34460  bj-opelid  34461  bj-ideqgALT  34463
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