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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 12992 | . 2 ⊢ (𝑀 ∈ 𝑉 → ((#‘𝑀) ∈ ℕ0 ∨ (#‘𝑀) = +∞)) | |
2 | elnn0 11171 | . . . 4 ⊢ ((#‘𝑀) ∈ ℕ0 ↔ ((#‘𝑀) ∈ ℕ ∨ (#‘𝑀) = 0)) | |
3 | exmidne 2792 | . . . . . . . 8 ⊢ ((#‘𝑀) = 1 ∨ (#‘𝑀) ≠ 1) | |
4 | nngt1ne1 10924 | . . . . . . . . 9 ⊢ ((#‘𝑀) ∈ ℕ → (1 < (#‘𝑀) ↔ (#‘𝑀) ≠ 1)) | |
5 | 4 | orbi2d 734 | . . . . . . . 8 ⊢ ((#‘𝑀) ∈ ℕ → (((#‘𝑀) = 1 ∨ 1 < (#‘𝑀)) ↔ ((#‘𝑀) = 1 ∨ (#‘𝑀) ≠ 1))) |
6 | 3, 5 | mpbiri 247 | . . . . . . 7 ⊢ ((#‘𝑀) ∈ ℕ → ((#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
7 | 6 | olcd 407 | . . . . . 6 ⊢ ((#‘𝑀) ∈ ℕ → ((#‘𝑀) = 0 ∨ ((#‘𝑀) = 1 ∨ 1 < (#‘𝑀)))) |
8 | 3orass 1034 | . . . . . 6 ⊢ (((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀)) ↔ ((#‘𝑀) = 0 ∨ ((#‘𝑀) = 1 ∨ 1 < (#‘𝑀)))) | |
9 | 7, 8 | sylibr 223 | . . . . 5 ⊢ ((#‘𝑀) ∈ ℕ → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
10 | 3mix1 1223 | . . . . 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 206 | . . 3 ⊢ ((#‘𝑀) ∈ ℕ0 → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀))) |
13 | 1re 9918 | . . . . . 6 ⊢ 1 ∈ ℝ | |
14 | ltpnf 11830 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
16 | breq2 4587 | . . . . 5 ⊢ ((#‘𝑀) = +∞ → (1 < (#‘𝑀) ↔ 1 < +∞)) | |
17 | 15, 16 | mpbiri 247 | . . . 4 ⊢ ((#‘𝑀) = +∞ → 1 < (#‘𝑀)) |
18 | 17 | 3mix3d 1231 | . . 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 1030 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 < clt 9953 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-wrecs 7294 df-recs 7355 df-rdg 7393 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-xnn0 11241 df-z 11255 df-uz 11564 df-hash 12980 |
This theorem is referenced by: hashge2el2difr 13118 01eq0ring 19093 tgldimor 25197 frgrawopreg 26576 frgrwopreg 41486 |
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