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Mirrors > Home > MPE Home > Th. List > negn0 | Structured version Visualization version GIF version |
Description: The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
negn0 | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4376 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | ssel 4002 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ)) | |
3 | renegcl 11599 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
4 | negeq 11528 | . . . . . . . . . . . 12 ⊢ (𝑧 = -𝑥 → -𝑧 = --𝑥) | |
5 | 4 | eleq1d 2829 | . . . . . . . . . . 11 ⊢ (𝑧 = -𝑥 → (-𝑧 ∈ 𝐴 ↔ --𝑥 ∈ 𝐴)) |
6 | 5 | elrab3 3709 | . . . . . . . . . 10 ⊢ (-𝑥 ∈ ℝ → (-𝑥 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ --𝑥 ∈ 𝐴)) |
7 | 3, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (-𝑥 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ --𝑥 ∈ 𝐴)) |
8 | recn 11274 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
9 | 8 | negnegd 11638 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → --𝑥 = 𝑥) |
10 | 9 | eleq1d 2829 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (--𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
11 | 7, 10 | bitrd 279 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (-𝑥 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)) |
12 | 11 | biimprd 248 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ 𝐴 → -𝑥 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴})) |
13 | 2, 12 | syli 39 | . . . . . 6 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → -𝑥 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴})) |
14 | elex2 2821 | . . . . . 6 ⊢ (-𝑥 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) | |
15 | 13, 14 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴})) |
16 | n0 4376 | . . . . 5 ⊢ ({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅ ↔ ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) | |
17 | 15, 16 | imbitrrdi 252 | . . . 4 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅)) |
18 | 17 | exlimdv 1932 | . . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 𝑥 ∈ 𝐴 → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅)) |
19 | 1, 18 | biimtrid 242 | . 2 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅)) |
20 | 19 | imp 406 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ≠ wne 2946 {crab 3443 ⊆ wss 3976 ∅c0 4352 ℝcr 11183 -cneg 11521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-ltxr 11329 df-sub 11522 df-neg 11523 |
This theorem is referenced by: supminf 13000 supminfxr 45379 |
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