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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 8750, 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 8750 | . 2 ⊢ (⊤ → 𝑅 Er 𝐴) |
10 | 9 | mptru 1541 | 1 ⊢ 𝑅 Er 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ⊤wtru 1535 ∈ wcel 2099 class class class wbr 5148 Rel wrel 5683 Er wer 8721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-er 8724 |
This theorem is referenced by: eqer 8759 0er 8761 ecopover 8839 ener 9021 gicer 19230 hmpher 23687 phtpcer 24920 vitalilem1 25536 tgjustf 28276 erclwwlk 29832 erclwwlkn 29881 gricer 47190 |
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