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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 35977 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | |
2 | cosscnvelrels 35773 | . . . 4 ⊢ (𝑟 ∈ Rels → ≀ ◡𝑟 ∈ Rels ) | |
3 | 2 | biantrud 534 | . . 3 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ⊆ I ↔ ( ≀ ◡𝑟 ⊆ I ∧ ≀ ◡𝑟 ∈ Rels ))) |
4 | cosselcnvrefrels2 35810 | . . 3 ⊢ ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ( ≀ ◡𝑟 ⊆ I ∧ ≀ ◡𝑟 ∈ Rels )) | |
5 | 3, 4 | syl6rbbr 292 | . 2 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑟 ⊆ I )) |
6 | 1, 5 | rabimbieq 35549 | 1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3141 ⊆ wss 3933 I cid 5456 ◡ccnv 5551 ≀ ccoss 35489 Rels crels 35491 CnvRefRels ccnvrefrels 35497 Disjs cdisjs 35522 |
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 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-coss 35695 df-rels 35761 df-ssr 35774 df-cnvrefs 35799 df-cnvrefrels 35800 df-disjss 35972 df-disjs 35973 |
This theorem is referenced by: dfdisjs3 35979 dfdisjs4 35980 dfdisjs5 35981 |
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