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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 4305 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | ssel 3936 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ)) | |
3 | renegcl 11461 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
4 | negeq 11390 | . . . . . . . . . . . 12 ⊢ (𝑧 = -𝑥 → -𝑧 = --𝑥) | |
5 | 4 | eleq1d 2822 | . . . . . . . . . . 11 ⊢ (𝑧 = -𝑥 → (-𝑧 ∈ 𝐴 ↔ --𝑥 ∈ 𝐴)) |
6 | 5 | elrab3 3645 | . . . . . . . . . 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 11138 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
9 | 8 | negnegd 11500 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → --𝑥 = 𝑥) |
10 | 9 | eleq1d 2822 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (--𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
11 | 7, 10 | bitrd 278 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (-𝑥 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)) |
12 | 11 | biimprd 247 | . . . . . . 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 2816 | . . . . . 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 4305 | . . . . 5 ⊢ ({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅ ↔ ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) | |
17 | 15, 16 | syl6ibr 251 | . . . 4 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅)) |
18 | 17 | exlimdv 1936 | . . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 𝑥 ∈ 𝐴 → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅)) |
19 | 1, 18 | biimtrid 241 | . 2 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅)) |
20 | 19 | imp 407 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2942 {crab 3406 ⊆ wss 3909 ∅c0 4281 ℝcr 11047 -cneg 11383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-po 5544 df-so 5545 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-ltxr 11191 df-sub 11384 df-neg 11385 |
This theorem is referenced by: supminf 12857 supminfxr 43673 |
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