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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfdisjs2 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
Ref | Expression |
---|---|
dfdisjs2 | ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisjs 37220 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | |
2 | cosselcnvrefrels2 37050 | . . 3 ⊢ ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ( ≀ ◡𝑟 ⊆ I ∧ ≀ ◡𝑟 ∈ Rels )) | |
3 | cosscnvelrels 37009 | . . . 4 ⊢ (𝑟 ∈ Rels → ≀ ◡𝑟 ∈ Rels ) | |
4 | 3 | biantrud 533 | . . 3 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ⊆ I ↔ ( ≀ ◡𝑟 ⊆ I ∧ ≀ ◡𝑟 ∈ Rels ))) |
5 | 2, 4 | bitr4id 290 | . 2 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑟 ⊆ I )) |
6 | 1, 5 | rabimbieq 36761 | 1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 ⊆ wss 3914 I cid 5534 ◡ccnv 5636 ≀ ccoss 36684 Rels crels 36686 CnvRefRels ccnvrefrels 36692 Disjs cdisjs 36717 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-coss 36923 df-rels 36997 df-ssr 37010 df-cnvrefs 37037 df-cnvrefrels 37038 df-disjss 37215 df-disjs 37216 |
This theorem is referenced by: dfdisjs3 37222 dfdisjs4 37223 dfdisjs5 37224 |
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