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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjs | Structured version Visualization version GIF version |
Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.) |
Ref | Expression |
---|---|
eldisjs | ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisjs 35974 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | |
2 | cnveq 5737 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
3 | 2 | cosseqd 35706 | . . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅) |
4 | 3 | eleq1d 2896 | . 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels )) |
5 | 1, 4 | rabeqel 35549 | 1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ◡ccnv 5547 ≀ ccoss 35486 Rels crels 35488 CnvRefRels ccnvrefrels 35494 Disjs cdisjs 35519 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-in 3936 df-ss 3945 df-br 5060 df-opab 5122 df-cnv 5556 df-coss 35692 df-disjss 35969 df-disjs 35970 |
This theorem is referenced by: eldisjs2 35989 eldisjsdisj 35993 |
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