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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 36624. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-rest10b | ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3949 | . . 3 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) ↔ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅})) | |
2 | 0ex 5302 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | elsn2 4663 | . . . . 5 ⊢ (𝑋 ∈ {∅} ↔ 𝑋 = ∅) |
4 | neqne 2938 | . . . . 5 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
5 | 3, 4 | sylnbi 329 | . . . 4 ⊢ (¬ 𝑋 ∈ {∅} → 𝑋 ≠ ∅) |
6 | 5 | anim2i 615 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅}) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅)) |
7 | 1, 6 | sylbi 216 | . 2 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅)) |
8 | bj-rest10 36624 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅})) | |
9 | 8 | imp 405 | . 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∖ cdif 3936 ∅c0 4318 {csn 4624 (class class class)co 7416 ↾t crest 17401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-rest 17403 |
This theorem is referenced by: (None) |
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