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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rest10b | Structured version Visualization version GIF version |
Description: Alternate version of bj-rest10 35255. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-rest10b | ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3902 | . . 3 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) ↔ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅})) | |
2 | 0ex 5235 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | elsn2 4606 | . . . . 5 ⊢ (𝑋 ∈ {∅} ↔ 𝑋 = ∅) |
4 | neqne 2953 | . . . . 5 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
5 | 3, 4 | sylnbi 330 | . . . 4 ⊢ (¬ 𝑋 ∈ {∅} → 𝑋 ≠ ∅) |
6 | 5 | anim2i 617 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅}) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅)) |
7 | 1, 6 | sylbi 216 | . 2 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅)) |
8 | bj-rest10 35255 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅})) | |
9 | 8 | imp 407 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅) → (𝑋 ↾t ∅) = {∅}) |
10 | 7, 9 | syl 17 | 1 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∖ cdif 3889 ∅c0 4262 {csn 4567 (class class class)co 7271 ↾t crest 17129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-rest 17131 |
This theorem is referenced by: (None) |
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