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Mirrors > Home > MPE Home > Th. List > tgldimor | Structured version Visualization version GIF version |
Description: Excluded-middle like statement allowing to treat dimension zero as a special case. (Contributed by Thierry Arnoux, 11-Apr-2019.) |
Ref | Expression |
---|---|
tgldimor.p | ⊢ 𝑃 = (𝐸‘𝐹) |
tgldimor.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
Ref | Expression |
---|---|
tgldimor | ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgldimor.p | . . . . . 6 ⊢ 𝑃 = (𝐸‘𝐹) | |
2 | fvex 6354 | . . . . . 6 ⊢ (𝐸‘𝐹) ∈ V | |
3 | 1, 2 | eqeltri 2827 | . . . . 5 ⊢ 𝑃 ∈ V |
4 | hashv01gt1 13319 | . . . . 5 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) |
6 | 3orass 1075 | . . . 4 ⊢ (((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)))) | |
7 | 5, 6 | mpbi 220 | . . 3 ⊢ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) |
8 | 1p1e2 11318 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
9 | 1z 11591 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
10 | nn0z 11584 | . . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℤ) | |
11 | zltp1le 11611 | . . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑃) ∈ ℤ) → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) | |
12 | 9, 10, 11 | sylancr 698 | . . . . . . . 8 ⊢ ((♯‘𝑃) ∈ ℕ0 → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) |
13 | 12 | biimpac 504 | . . . . . . 7 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → (1 + 1) ≤ (♯‘𝑃)) |
14 | 8, 13 | syl5eqbrr 4832 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → 2 ≤ (♯‘𝑃)) |
15 | 2re 11274 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
16 | 15 | rexri 10281 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
17 | pnfge 12149 | . . . . . . . . 9 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 ⊢ 2 ≤ +∞ |
19 | breq2 4800 | . . . . . . . 8 ⊢ ((♯‘𝑃) = +∞ → (2 ≤ (♯‘𝑃) ↔ 2 ≤ +∞)) | |
20 | 18, 19 | mpbiri 248 | . . . . . . 7 ⊢ ((♯‘𝑃) = +∞ → 2 ≤ (♯‘𝑃)) |
21 | 20 | adantl 473 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) = +∞) → 2 ≤ (♯‘𝑃)) |
22 | hashnn0pnf 13316 | . . . . . . 7 ⊢ (𝑃 ∈ V → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) | |
23 | 3, 22 | mp1i 13 | . . . . . 6 ⊢ (1 < (♯‘𝑃) → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) |
24 | 14, 21, 23 | mpjaodan 862 | . . . . 5 ⊢ (1 < (♯‘𝑃) → 2 ≤ (♯‘𝑃)) |
25 | 24 | orim2i 541 | . . . 4 ⊢ (((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
26 | 25 | orim2i 541 | . . 3 ⊢ (((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) → ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)))) |
27 | 7, 26 | mp1i 13 | . 2 ⊢ (𝜑 → ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)))) |
28 | tgldimor.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
29 | ne0i 4056 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → 𝑃 ≠ ∅) | |
30 | hasheq0 13338 | . . . . . . 7 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ↔ 𝑃 = ∅)) | |
31 | 3, 30 | ax-mp 5 | . . . . . 6 ⊢ ((♯‘𝑃) = 0 ↔ 𝑃 = ∅) |
32 | 31 | biimpi 206 | . . . . 5 ⊢ ((♯‘𝑃) = 0 → 𝑃 = ∅) |
33 | 32 | necon3ai 2949 | . . . 4 ⊢ (𝑃 ≠ ∅ → ¬ (♯‘𝑃) = 0) |
34 | 28, 29, 33 | 3syl 18 | . . 3 ⊢ (𝜑 → ¬ (♯‘𝑃) = 0) |
35 | biorf 419 | . . 3 ⊢ (¬ (♯‘𝑃) = 0 → (((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))))) | |
36 | 34, 35 | syl 17 | . 2 ⊢ (𝜑 → (((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))))) |
37 | 27, 36 | mpbird 247 | 1 ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 ∨ w3o 1071 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 Vcvv 3332 ∅c0 4050 class class class wbr 4796 ‘cfv 6041 (class class class)co 6805 0cc0 10120 1c1 10121 + caddc 10123 +∞cpnf 10255 ℝ*cxr 10257 < clt 10258 ≤ cle 10259 2c2 11254 ℕ0cn0 11476 ℤcz 11561 ♯chash 13303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-card 8947 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-n0 11477 df-xnn0 11548 df-z 11562 df-uz 11872 df-fz 12512 df-hash 13304 |
This theorem is referenced by: tgifscgr 25594 tgcgrxfr 25604 tgbtwnconn3 25663 legtrid 25677 hpgerlem 25848 |
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