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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 37089. (Contributed by BJ, 27-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| bj-rest10b | ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eldif 3961 | . . 3 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) ↔ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅})) | |
| 2 | 0ex 5307 | . . . . . 6 ⊢ ∅ ∈ V | |
| 3 | 2 | elsn2 4665 | . . . . 5 ⊢ (𝑋 ∈ {∅} ↔ 𝑋 = ∅) | 
| 4 | neqne 2948 | . . . . 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 217 | . 2 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅)) | 
| 8 | bj-rest10 37089 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅})) | |
| 9 | 8 | imp 406 | . 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 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∅c0 4333 {csn 4626 (class class class)co 7431 ↾t crest 17465 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rest 17467 | 
| This theorem is referenced by: (None) | 
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