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| Mirrors > Home > MPE Home > Th. List > Mathboxes > relsset | Structured version Visualization version GIF version | ||
| Description: The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.) | 
| Ref | Expression | 
|---|---|
| relsset | ⊢ Rel SSet | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-sset 35858 | . . 3 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) | |
| 2 | difss 4135 | . . 3 ⊢ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ⊆ (V × V) | |
| 3 | 1, 2 | eqsstri 4029 | . 2 ⊢ SSet ⊆ (V × V) | 
| 4 | df-rel 5691 | . 2 ⊢ (Rel SSet ↔ SSet ⊆ (V × V)) | |
| 5 | 3, 4 | mpbir 231 | 1 ⊢ Rel SSet | 
| Colors of variables: wff setvar class | 
| Syntax hints: Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 E cep 5582 × cxp 5682 ran crn 5685 Rel wrel 5689 ⊗ ctxp 35832 SSet csset 35834 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-ss 3967 df-rel 5691 df-sset 35858 | 
| This theorem is referenced by: brsset 35891 idsset 35892 | 
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