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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 35259. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-rest10b | ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3897 | . . 3 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) ↔ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅})) | |
2 | 0ex 5231 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | elsn2 4600 | . . . . 5 ⊢ (𝑋 ∈ {∅} ↔ 𝑋 = ∅) |
4 | neqne 2951 | . . . . 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 35259 | . . 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 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ∅c0 4256 {csn 4561 (class class class)co 7275 ↾t crest 17131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-rest 17133 |
This theorem is referenced by: (None) |
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