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Mirrors > Home > MPE Home > Th. List > Mathboxes > resnonrel | Structured version Visualization version GIF version |
Description: A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
Ref | Expression |
---|---|
resnonrel | ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3997 | . . . 4 ⊢ 𝐵 ⊆ V | |
2 | ssres2 6004 | . . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V) |
4 | cnvnonrel 43083 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
5 | 4 | cnveqi 5871 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅ |
6 | cnvcnv2 6192 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V) | |
7 | cnv0 6140 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 5, 6, 7 | 3eqtr3i 2761 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅ |
9 | 3, 8 | sseqtri 4009 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ |
10 | ss0b 4393 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅) | |
11 | 9, 10 | mpbi 229 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3463 ∖ cdif 3936 ⊆ wss 3939 ∅c0 4318 ◡ccnv 5671 ↾ cres 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 df-res 5684 |
This theorem is referenced by: imanonrel 43088 |
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