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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 13891 | . 2 ⊢ (𝑀 ∈ 𝑉 → ((♯‘𝑀) ∈ ℕ0 ∨ (♯‘𝑀) = +∞)) | |
2 | elnn0 12075 | . . . 4 ⊢ ((♯‘𝑀) ∈ ℕ0 ↔ ((♯‘𝑀) ∈ ℕ ∨ (♯‘𝑀) = 0)) | |
3 | exmidne 2945 | . . . . . . . 8 ⊢ ((♯‘𝑀) = 1 ∨ (♯‘𝑀) ≠ 1) | |
4 | nngt1ne1 11842 | . . . . . . . . 9 ⊢ ((♯‘𝑀) ∈ ℕ → (1 < (♯‘𝑀) ↔ (♯‘𝑀) ≠ 1)) | |
5 | 4 | orbi2d 916 | . . . . . . . 8 ⊢ ((♯‘𝑀) ∈ ℕ → (((♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)) ↔ ((♯‘𝑀) = 1 ∨ (♯‘𝑀) ≠ 1))) |
6 | 3, 5 | mpbiri 261 | . . . . . . 7 ⊢ ((♯‘𝑀) ∈ ℕ → ((♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
7 | 6 | olcd 874 | . . . . . 6 ⊢ ((♯‘𝑀) ∈ ℕ → ((♯‘𝑀) = 0 ∨ ((♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)))) |
8 | 3orass 1092 | . . . . . 6 ⊢ (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)) ↔ ((♯‘𝑀) = 0 ∨ ((♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)))) | |
9 | 7, 8 | sylibr 237 | . . . . 5 ⊢ ((♯‘𝑀) ∈ ℕ → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
10 | 3mix1 1332 | . . . . 5 ⊢ ((♯‘𝑀) = 0 → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) | |
11 | 9, 10 | jaoi 857 | . . . 4 ⊢ (((♯‘𝑀) ∈ ℕ ∨ (♯‘𝑀) = 0) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
12 | 2, 11 | sylbi 220 | . . 3 ⊢ ((♯‘𝑀) ∈ ℕ0 → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
13 | 1re 10816 | . . . . . 6 ⊢ 1 ∈ ℝ | |
14 | ltpnf 12695 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
16 | breq2 5047 | . . . . 5 ⊢ ((♯‘𝑀) = +∞ → (1 < (♯‘𝑀) ↔ 1 < +∞)) | |
17 | 15, 16 | mpbiri 261 | . . . 4 ⊢ ((♯‘𝑀) = +∞ → 1 < (♯‘𝑀)) |
18 | 17 | 3mix3d 1340 | . . 3 ⊢ ((♯‘𝑀) = +∞ → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
19 | 12, 18 | jaoi 857 | . 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 847 ∨ w3o 1088 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 class class class wbr 5043 ‘cfv 6369 ℝcr 10711 0cc0 10712 1c1 10713 +∞cpnf 10847 < clt 10850 ℕcn 11813 ℕ0cn0 12073 ♯chash 13879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-hash 13880 |
This theorem is referenced by: hashge2el2difr 14030 symgvalstruct 18761 01eq0ring 20282 tgldimor 26565 frgrwopreg 28378 |
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