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Mirrors > Home > MPE Home > Th. List > hashnn0n0nn | Structured version Visualization version GIF version |
Description: If a nonnegative integer is the size of a set which contains at least one element, this integer is a positive integer. (Contributed by Alexander van der Vekens, 9-Jan-2018.) |
Ref | Expression |
---|---|
hashnn0n0nn | ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) ∧ ((#‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉)) → 𝑌 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 3880 | . . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → 𝑉 ≠ ∅) | |
2 | hashge1 13039 | . . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑉 ≠ ∅) → 1 ≤ (#‘𝑉)) | |
3 | 1, 2 | sylan2 490 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 1 ≤ (#‘𝑉)) |
4 | simpr 476 | . . . . . . . . 9 ⊢ ((1 ≤ (#‘𝑉) ∧ (#‘𝑉) ∈ ℕ0) → (#‘𝑉) ∈ ℕ0) | |
5 | 0lt1 10429 | . . . . . . . . . . . . 13 ⊢ 0 < 1 | |
6 | 0re 9919 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ | |
7 | 1re 9918 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ | |
8 | 6, 7 | ltnlei 10037 | . . . . . . . . . . . . 13 ⊢ (0 < 1 ↔ ¬ 1 ≤ 0) |
9 | 5, 8 | mpbi 219 | . . . . . . . . . . . 12 ⊢ ¬ 1 ≤ 0 |
10 | breq2 4587 | . . . . . . . . . . . 12 ⊢ ((#‘𝑉) = 0 → (1 ≤ (#‘𝑉) ↔ 1 ≤ 0)) | |
11 | 9, 10 | mtbiri 316 | . . . . . . . . . . 11 ⊢ ((#‘𝑉) = 0 → ¬ 1 ≤ (#‘𝑉)) |
12 | 11 | necon2ai 2811 | . . . . . . . . . 10 ⊢ (1 ≤ (#‘𝑉) → (#‘𝑉) ≠ 0) |
13 | 12 | adantr 480 | . . . . . . . . 9 ⊢ ((1 ≤ (#‘𝑉) ∧ (#‘𝑉) ∈ ℕ0) → (#‘𝑉) ≠ 0) |
14 | elnnne0 11183 | . . . . . . . . 9 ⊢ ((#‘𝑉) ∈ ℕ ↔ ((#‘𝑉) ∈ ℕ0 ∧ (#‘𝑉) ≠ 0)) | |
15 | 4, 13, 14 | sylanbrc 695 | . . . . . . . 8 ⊢ ((1 ≤ (#‘𝑉) ∧ (#‘𝑉) ∈ ℕ0) → (#‘𝑉) ∈ ℕ) |
16 | 15 | ex 449 | . . . . . . 7 ⊢ (1 ≤ (#‘𝑉) → ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℕ)) |
17 | 3, 16 | syl 17 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℕ)) |
18 | 17 | impancom 455 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) ∈ ℕ0) → (𝑁 ∈ 𝑉 → (#‘𝑉) ∈ ℕ)) |
19 | 18 | com12 32 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) ∈ ℕ0) → (#‘𝑉) ∈ ℕ)) |
20 | eleq1 2676 | . . . . . 6 ⊢ ((#‘𝑉) = 𝑌 → ((#‘𝑉) ∈ ℕ0 ↔ 𝑌 ∈ ℕ0)) | |
21 | 20 | anbi2d 736 | . . . . 5 ⊢ ((#‘𝑉) = 𝑌 → ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) ∈ ℕ0) ↔ (𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0))) |
22 | eleq1 2676 | . . . . 5 ⊢ ((#‘𝑉) = 𝑌 → ((#‘𝑉) ∈ ℕ ↔ 𝑌 ∈ ℕ)) | |
23 | 21, 22 | imbi12d 333 | . . . 4 ⊢ ((#‘𝑉) = 𝑌 → (((𝑉 ∈ 𝑊 ∧ (#‘𝑉) ∈ ℕ0) → (#‘𝑉) ∈ ℕ) ↔ ((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) → 𝑌 ∈ ℕ))) |
24 | 19, 23 | syl5ib 233 | . . 3 ⊢ ((#‘𝑉) = 𝑌 → (𝑁 ∈ 𝑉 → ((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) → 𝑌 ∈ ℕ))) |
25 | 24 | imp 444 | . 2 ⊢ (((#‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉) → ((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) → 𝑌 ∈ ℕ)) |
26 | 25 | impcom 445 | 1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) ∧ ((#‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉)) → 𝑌 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 0cc0 9815 1c1 9816 < clt 9953 ≤ cle 9954 ℕcn 10897 ℕ0cn0 11169 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: cusgrasize2inds 26005 cusgrsize2inds 40669 |
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