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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 36101 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | |
2 | cnveq 5708 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
3 | 2 | cosseqd 35833 | . . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅) |
4 | 3 | eleq1d 2874 | . 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels )) |
5 | 1, 4 | rabeqel 35676 | 1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ◡ccnv 5518 ≀ ccoss 35613 Rels crels 35615 CnvRefRels ccnvrefrels 35621 Disjs cdisjs 35646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-br 5031 df-opab 5093 df-cnv 5527 df-coss 35819 df-disjss 36096 df-disjs 36097 |
This theorem is referenced by: eldisjs2 36116 eldisjsdisj 36120 |
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