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Mirrors > Home > MPE Home > Th. List > hashgt0elex | Structured version Visualization version GIF version |
Description: If the size of a set is greater than zero, then the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.) |
Ref | Expression |
---|---|
hashgt0elex | ⊢ ((𝑉 ∈ 𝑊 ∧ 0 < (♯‘𝑉)) → ∃𝑥 𝑥 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1773 | . . . . . . . 8 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑉 ↔ ¬ ∃𝑥 𝑥 ∈ 𝑉) | |
2 | eq0 4307 | . . . . . . . . . 10 ⊢ (𝑉 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝑉) | |
3 | 2 | biimpri 229 | . . . . . . . . 9 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑉 → 𝑉 = ∅) |
4 | 3 | a1d 25 | . . . . . . . 8 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑉 → (𝑉 ∈ 𝑊 → 𝑉 = ∅)) |
5 | 1, 4 | sylbir 236 | . . . . . . 7 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝑉 → (𝑉 ∈ 𝑊 → 𝑉 = ∅)) |
6 | 5 | impcom 408 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ ∃𝑥 𝑥 ∈ 𝑉) → 𝑉 = ∅) |
7 | hashle00 13751 | . . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) ≤ 0 ↔ 𝑉 = ∅)) | |
8 | 7 | adantr 481 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ ∃𝑥 𝑥 ∈ 𝑉) → ((♯‘𝑉) ≤ 0 ↔ 𝑉 = ∅)) |
9 | 6, 8 | mpbird 258 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ ∃𝑥 𝑥 ∈ 𝑉) → (♯‘𝑉) ≤ 0) |
10 | hashxrcl 13708 | . . . . . . . 8 ⊢ (𝑉 ∈ 𝑊 → (♯‘𝑉) ∈ ℝ*) | |
11 | 0xr 10677 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
12 | xrlenlt 10695 | . . . . . . . 8 ⊢ (((♯‘𝑉) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((♯‘𝑉) ≤ 0 ↔ ¬ 0 < (♯‘𝑉))) | |
13 | 10, 11, 12 | sylancl 586 | . . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) ≤ 0 ↔ ¬ 0 < (♯‘𝑉))) |
14 | 13 | bicomd 224 | . . . . . 6 ⊢ (𝑉 ∈ 𝑊 → (¬ 0 < (♯‘𝑉) ↔ (♯‘𝑉) ≤ 0)) |
15 | 14 | adantr 481 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ ∃𝑥 𝑥 ∈ 𝑉) → (¬ 0 < (♯‘𝑉) ↔ (♯‘𝑉) ≤ 0)) |
16 | 9, 15 | mpbird 258 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ ∃𝑥 𝑥 ∈ 𝑉) → ¬ 0 < (♯‘𝑉)) |
17 | 16 | ex 413 | . . 3 ⊢ (𝑉 ∈ 𝑊 → (¬ ∃𝑥 𝑥 ∈ 𝑉 → ¬ 0 < (♯‘𝑉))) |
18 | 17 | con4d 115 | . 2 ⊢ (𝑉 ∈ 𝑊 → (0 < (♯‘𝑉) → ∃𝑥 𝑥 ∈ 𝑉)) |
19 | 18 | imp 407 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 0 < (♯‘𝑉)) → ∃𝑥 𝑥 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∅c0 4290 class class class wbr 5058 ‘cfv 6349 0cc0 10526 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 ♯chash 13680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-xnn0 11957 df-z 11971 df-uz 12233 df-fz 12883 df-hash 13681 |
This theorem is referenced by: hashgt0elexb 13753 hashgt23el 13775 fi1uzind 13845 brfi1indALT 13848 |
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