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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfdisjALTV2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV2 38689. (Contributed by Peter Mazsa, 27-Jul-2021.) |
| Ref | Expression |
|---|---|
| dfdisjALTV2 | ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-disjALTV 38706 | . 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | |
| 2 | cnvrefrelcoss2 38538 | . . 3 ⊢ ( CnvRefRel ≀ ◡𝑅 ↔ ≀ ◡𝑅 ⊆ I ) | |
| 3 | 2 | anbi1i 624 | . 2 ⊢ (( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅) ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) |
| 4 | 1, 3 | bitri 275 | 1 ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ⊆ wss 3951 I cid 5577 ◡ccnv 5684 Rel wrel 5690 ≀ ccoss 38182 CnvRefRel wcnvrefrel 38191 Disj wdisjALTV 38216 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-coss 38412 df-cnvrefrel 38528 df-disjALTV 38706 |
| This theorem is referenced by: dfdisjALTV3 38716 dfdisjALTV4 38717 dfdisjALTV5 38718 dfeldisj2 38719 disjxrn 38747 disjorimxrn 38749 disjALTVid 38756 |
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