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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 37464. (Contributed by Peter Mazsa, 27-Jul-2021.) |
Ref | Expression |
---|---|
dfdisjALTV2 | ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-disjALTV 37481 | . 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | |
2 | cnvrefrelcoss2 37313 | . . 3 ⊢ ( CnvRefRel ≀ ◡𝑅 ↔ ≀ ◡𝑅 ⊆ I ) | |
3 | 2 | anbi1i 625 | . 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 205 ∧ wa 397 ⊆ wss 3946 I cid 5569 ◡ccnv 5671 Rel wrel 5677 ≀ ccoss 36949 CnvRefRel wcnvrefrel 36958 Disj wdisjALTV 36983 |
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 5295 ax-nul 5302 ax-pr 5423 |
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 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-coss 37187 df-cnvrefrel 37303 df-disjALTV 37481 |
This theorem is referenced by: dfdisjALTV3 37491 dfdisjALTV4 37492 dfdisjALTV5 37493 dfeldisj2 37494 disjxrn 37522 disjorimxrn 37524 disjALTVid 37531 |
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