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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 8675, 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 483 | . . 3 ⊢ ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) |
5 | iseri.3 | . . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) | |
6 | 5 | adantl 483 | . . 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 8675 | . 2 ⊢ (⊤ → 𝑅 Er 𝐴) |
10 | 9 | mptru 1549 | 1 ⊢ 𝑅 Er 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ⊤wtru 1543 ∈ wcel 2107 class class class wbr 5106 Rel wrel 5639 Er wer 8646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-er 8649 |
This theorem is referenced by: eqer 8684 0er 8686 ecopover 8761 ener 8942 gicer 19067 hmpher 23138 phtpcer 24361 vitalilem1 24975 tgjustf 27418 erclwwlk 28970 erclwwlkn 29019 |
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