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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelidb | Structured version Visualization version GIF version |
Description: Characterization of the
ordered pair elements of the identity relation.
Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than ⊤ which already appears in the proof. Here for instance this could be the definition I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.) |
Ref | Expression |
---|---|
bj-opelidb | ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-id 5453 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) |
3 | eqeq12 2834 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵)) | |
4 | 3 | adantl 484 | . . 3 ⊢ ((⊤ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵)) |
5 | 2, 4 | opelopabbv 34463 | . 2 ⊢ (⊤ → (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))) |
6 | 5 | mptru 1543 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 Vcvv 3491 〈cop 4566 {copab 5121 I cid 5452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-id 5453 |
This theorem is referenced by: bj-opelidb1 34473 bj-opelid 34476 |
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