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Mirrors > Home > MPE Home > Th. List > iseri | Structured version Visualization version GIF version |
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8726, which avoids the need to provide a "dummy antecedent" 𝜑 if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
iseri.1 | ⊢ Rel 𝑅 |
iseri.2 | ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥) |
iseri.3 | ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
iseri.4 | ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥) |
Ref | Expression |
---|---|
iseri | ⊢ 𝑅 Er 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseri.1 | . . . 4 ⊢ Rel 𝑅 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Rel 𝑅) |
3 | iseri.2 | . . . 4 ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) |
5 | iseri.3 | . . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) | |
6 | 5 | adantl 481 | . . 3 ⊢ ((⊤ ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) |
7 | iseri.4 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥) | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) |
9 | 2, 4, 6, 8 | iserd 8726 | . 2 ⊢ (⊤ → 𝑅 Er 𝐴) |
10 | 9 | mptru 1540 | 1 ⊢ 𝑅 Er 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ⊤wtru 1534 ∈ wcel 2098 class class class wbr 5139 Rel wrel 5672 Er wer 8697 |
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-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
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-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-er 8700 |
This theorem is referenced by: eqer 8735 0er 8737 ecopover 8812 ener 8994 gicer 19198 hmpher 23632 phtpcer 24865 vitalilem1 25481 tgjustf 28217 erclwwlk 29770 erclwwlkn 29819 |
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