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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 5460 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) |
3 | eqeq12 2835 | . . . 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 34438 | . 2 ⊢ (⊤ → (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))) |
6 | 5 | mptru 1544 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 Vcvv 3494 〈cop 4573 {copab 5128 I cid 5459 |
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 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-id 5460 |
This theorem is referenced by: bj-opelidb1 34448 bj-opelid 34451 |
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