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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelidb1ALT | Structured version Visualization version GIF version |
Description: Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-opelidb1ALT | ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5139 | . . 3 ⊢ (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I ) | |
2 | reli 5816 | . . . 4 ⊢ Rel I | |
3 | 2 | brrelex1i 5722 | . . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V) |
4 | 1, 3 | sylbir 234 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I → 𝐴 ∈ V) |
5 | inex1g 5309 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V) | |
6 | bj-opelid 36527 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵)) |
8 | 4, 7 | biadanii 819 | 1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∩ cin 3939 ⟨cop 4626 class class class wbr 5138 I cid 5563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 |
This theorem is referenced by: (None) |
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