![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rest10b | Structured version Visualization version GIF version |
Description: Alternate version of bj-rest10 33563. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-rest10b | ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3808 | . . 3 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) ↔ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅})) | |
2 | 0ex 5016 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | elsn2 4434 | . . . . 5 ⊢ (𝑋 ∈ {∅} ↔ 𝑋 = ∅) |
4 | neqne 3007 | . . . . 5 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
5 | 3, 4 | sylnbi 322 | . . . 4 ⊢ (¬ 𝑋 ∈ {∅} → 𝑋 ≠ ∅) |
6 | 5 | anim2i 610 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ {∅}) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅)) |
7 | 1, 6 | sylbi 209 | . 2 ⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅)) |
8 | bj-rest10 33563 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅})) | |
9 | 8 | imp 397 | . 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 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∖ cdif 3795 ∅c0 4146 {csn 4399 (class class class)co 6910 ↾t crest 16441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-rest 16443 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |