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Mirrors > Home > MPE Home > Th. List > hashv01gt1 | Structured version Visualization version GIF version |
Description: The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.) |
Ref | Expression |
---|---|
hashv01gt1 | ⊢ (𝑀 ∈ 𝑉 → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnn0pnf 13170 | . 2 ⊢ (𝑀 ∈ 𝑉 → ((#‘𝑀) ∈ ℕ0 ∨ (#‘𝑀) = +∞)) | |
2 | elnn0 11332 | . . . 4 ⊢ ((#‘𝑀) ∈ ℕ0 ↔ ((#‘𝑀) ∈ ℕ ∨ (#‘𝑀) = 0)) | |
3 | exmidne 2833 | . . . . . . . 8 ⊢ ((#‘𝑀) = 1 ∨ (#‘𝑀) ≠ 1) | |
4 | nngt1ne1 11085 | . . . . . . . . 9 ⊢ ((#‘𝑀) ∈ ℕ → (1 < (#‘𝑀) ↔ (#‘𝑀) ≠ 1)) | |
5 | 4 | orbi2d 738 | . . . . . . . 8 ⊢ ((#‘𝑀) ∈ ℕ → (((#‘𝑀) = 1 ∨ 1 < (#‘𝑀)) ↔ ((#‘𝑀) = 1 ∨ (#‘𝑀) ≠ 1))) |
6 | 3, 5 | mpbiri 248 | . . . . . . 7 ⊢ ((#‘𝑀) ∈ ℕ → ((#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
7 | 6 | olcd 407 | . . . . . 6 ⊢ ((#‘𝑀) ∈ ℕ → ((#‘𝑀) = 0 ∨ ((#‘𝑀) = 1 ∨ 1 < (#‘𝑀)))) |
8 | 3orass 1057 | . . . . . 6 ⊢ (((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀)) ↔ ((#‘𝑀) = 0 ∨ ((#‘𝑀) = 1 ∨ 1 < (#‘𝑀)))) | |
9 | 7, 8 | sylibr 224 | . . . . 5 ⊢ ((#‘𝑀) ∈ ℕ → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
10 | 3mix1 1250 | . . . . 5 ⊢ ((#‘𝑀) = 0 → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) | |
11 | 9, 10 | jaoi 393 | . . . 4 ⊢ (((#‘𝑀) ∈ ℕ ∨ (#‘𝑀) = 0) → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
12 | 2, 11 | sylbi 207 | . . 3 ⊢ ((#‘𝑀) ∈ ℕ0 → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
13 | 1re 10077 | . . . . . 6 ⊢ 1 ∈ ℝ | |
14 | ltpnf 11992 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
16 | breq2 4689 | . . . . 5 ⊢ ((#‘𝑀) = +∞ → (1 < (#‘𝑀) ↔ 1 < +∞)) | |
17 | 15, 16 | mpbiri 248 | . . . 4 ⊢ ((#‘𝑀) = +∞ → 1 < (#‘𝑀)) |
18 | 17 | 3mix3d 1258 | . . 3 ⊢ ((#‘𝑀) = +∞ → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
19 | 12, 18 | jaoi 393 | . 2 ⊢ (((#‘𝑀) ∈ ℕ0 ∨ (#‘𝑀) = +∞) → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝑀 ∈ 𝑉 → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∨ w3o 1053 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ‘cfv 5926 ℝcr 9973 0cc0 9974 1c1 9975 +∞cpnf 10109 < clt 10112 ℕcn 11058 ℕ0cn0 11330 #chash 13157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-hash 13158 |
This theorem is referenced by: hashge2el2difr 13301 01eq0ring 19320 tgldimor 25442 frgrwopreg 27303 |
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