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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 3994 | . . . 4 ⊢ 𝐵 ⊆ V | |
2 | ssres2 5884 | . . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V) |
4 | cnvnonrel 39954 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
5 | 4 | cnveqi 5748 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅ |
6 | cnvcnv2 6053 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V) | |
7 | cnv0 6002 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 5, 6, 7 | 3eqtr3i 2855 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅ |
9 | 3, 8 | sseqtri 4006 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ |
10 | ss0b 4354 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅) | |
11 | 9, 10 | mpbi 232 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 Vcvv 3497 ∖ cdif 3936 ⊆ wss 3939 ∅c0 4294 ◡ccnv 5557 ↾ cres 5560 |
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 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-res 5570 |
This theorem is referenced by: imanonrel 39959 |
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