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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 36474 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | |
2 | cnveq 5726 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
3 | 2 | cosseqd 36206 | . . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅) |
4 | 3 | eleq1d 2818 | . 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels )) |
5 | 1, 4 | rabeqel 36049 | 1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ◡ccnv 5534 ≀ ccoss 35988 Rels crels 35990 CnvRefRels ccnvrefrels 35996 Disjs cdisjs 36021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-in 3860 df-ss 3870 df-br 5041 df-opab 5103 df-cnv 5543 df-coss 36192 df-disjss 36469 df-disjs 36470 |
This theorem is referenced by: eldisjs2 36489 eldisjsdisj 36493 |
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