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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 36101 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | |
2 | cosselcnvrefrels2 35934 | . . 3 ⊢ ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ( ≀ ◡𝑟 ⊆ I ∧ ≀ ◡𝑟 ∈ Rels )) | |
3 | cosscnvelrels 35897 | . . . 4 ⊢ (𝑟 ∈ Rels → ≀ ◡𝑟 ∈ Rels ) | |
4 | 3 | biantrud 535 | . . 3 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ⊆ I ↔ ( ≀ ◡𝑟 ⊆ I ∧ ≀ ◡𝑟 ∈ Rels ))) |
5 | 2, 4 | bitr4id 293 | . 2 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑟 ⊆ I )) |
6 | 1, 5 | rabimbieq 35673 | 1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 I cid 5424 ◡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 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-coss 35819 df-rels 35885 df-ssr 35898 df-cnvrefs 35923 df-cnvrefrels 35924 df-disjss 36096 df-disjs 36097 |
This theorem is referenced by: dfdisjs3 36103 dfdisjs4 36104 dfdisjs5 36105 |
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