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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inexeqex | Structured version Visualization version GIF version |
Description: Lemma for bj-opelid 34451 (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.) |
Ref | Expression |
---|---|
bj-inexeqex | ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 4023 | . . . . 5 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | df-ss 3952 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
3 | 1, 2 | sylib 220 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) |
4 | eleq1 2900 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ 𝑉 ↔ 𝐴 ∈ 𝑉)) | |
5 | 4 | biimpac 481 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = 𝐴) → 𝐴 ∈ 𝑉) |
6 | 3, 5 | sylan2 594 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉) |
7 | 6 | elexd 3514 | . 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) |
8 | eqimss2 4024 | . . . . 5 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
9 | sseqin2 4192 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
10 | 8, 9 | sylib 220 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵) |
11 | eleq1 2900 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → ((𝐴 ∩ 𝐵) ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) | |
12 | 11 | biimpac 481 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = 𝐵) → 𝐵 ∈ 𝑉) |
13 | 10, 12 | sylan2 594 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉) |
14 | 13 | elexd 3514 | . 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
15 | 7, 14 | jca 514 | 1 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 |
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-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 |
This theorem is referenced by: bj-elsn0 34450 bj-opelid 34451 bj-ideqgALT 34453 |
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