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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-restsn0 | Structured version Visualization version GIF version |
Description: An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 34372 and bj-restsnss2 34374. TODO: this is restsn 21777. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsn0 | ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4349 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | bj-restsnss2 34374 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ⊆ 𝐴) → ({∅} ↾t 𝐴) = {∅}) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∅c0 4290 {csn 4566 (class class class)co 7155 ↾t crest 16693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-rest 16695 |
This theorem is referenced by: (None) |
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