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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 3625 | . . . 4 ⊢ 𝐵 ⊆ V | |
2 | ssres2 5425 | . . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V) |
4 | cnvnonrel 37894 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
5 | 4 | cnveqi 5297 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅ |
6 | cnvcnv2 5588 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V) | |
7 | cnv0 5535 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 5, 6, 7 | 3eqtr3i 2652 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅ |
9 | 3, 8 | sseqtri 3637 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ |
10 | ss0b 3973 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅) | |
11 | 9, 10 | mpbi 220 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ◡ccnv 5113 ↾ cres 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-res 5126 |
This theorem is referenced by: imanonrel 37899 |
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