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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 | 1 | fvexi 6861 | . . . . 5 ⊢ 𝑃 ∈ V |
3 | hashv01gt1 14252 | . . . . 5 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) |
5 | 3orass 1091 | . . . 4 ⊢ (((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)))) | |
6 | 4, 5 | mpbi 229 | . . 3 ⊢ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) |
7 | 1p1e2 12285 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
8 | 1z 12540 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
9 | nn0z 12531 | . . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℤ) | |
10 | zltp1le 12560 | . . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑃) ∈ ℤ) → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) | |
11 | 8, 9, 10 | sylancr 588 | . . . . . . . 8 ⊢ ((♯‘𝑃) ∈ ℕ0 → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) |
12 | 11 | biimpac 480 | . . . . . . 7 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → (1 + 1) ≤ (♯‘𝑃)) |
13 | 7, 12 | eqbrtrrid 5146 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → 2 ≤ (♯‘𝑃)) |
14 | 2re 12234 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
15 | 14 | rexri 11220 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
16 | pnfge 13058 | . . . . . . . . 9 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ 2 ≤ +∞ |
18 | breq2 5114 | . . . . . . . 8 ⊢ ((♯‘𝑃) = +∞ → (2 ≤ (♯‘𝑃) ↔ 2 ≤ +∞)) | |
19 | 17, 18 | mpbiri 258 | . . . . . . 7 ⊢ ((♯‘𝑃) = +∞ → 2 ≤ (♯‘𝑃)) |
20 | 19 | adantl 483 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) = +∞) → 2 ≤ (♯‘𝑃)) |
21 | hashnn0pnf 14249 | . . . . . . 7 ⊢ (𝑃 ∈ V → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) | |
22 | 2, 21 | mp1i 13 | . . . . . 6 ⊢ (1 < (♯‘𝑃) → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) |
23 | 13, 20, 22 | mpjaodan 958 | . . . . 5 ⊢ (1 < (♯‘𝑃) → 2 ≤ (♯‘𝑃)) |
24 | 23 | orim2i 910 | . . . 4 ⊢ (((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
25 | 24 | orim2i 910 | . . 3 ⊢ (((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) → ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)))) |
26 | 6, 25 | mp1i 13 | . 2 ⊢ (𝜑 → ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)))) |
27 | tgldimor.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
28 | ne0i 4299 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → 𝑃 ≠ ∅) | |
29 | hasheq0 14270 | . . . . . . 7 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ↔ 𝑃 = ∅)) | |
30 | 2, 29 | ax-mp 5 | . . . . . 6 ⊢ ((♯‘𝑃) = 0 ↔ 𝑃 = ∅) |
31 | 30 | biimpi 215 | . . . . 5 ⊢ ((♯‘𝑃) = 0 → 𝑃 = ∅) |
32 | 31 | necon3ai 2969 | . . . 4 ⊢ (𝑃 ≠ ∅ → ¬ (♯‘𝑃) = 0) |
33 | 27, 28, 32 | 3syl 18 | . . 3 ⊢ (𝜑 → ¬ (♯‘𝑃) = 0) |
34 | biorf 936 | . . 3 ⊢ (¬ (♯‘𝑃) = 0 → (((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))))) | |
35 | 33, 34 | syl 17 | . 2 ⊢ (𝜑 → (((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))))) |
36 | 26, 35 | mpbird 257 | 1 ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ∅c0 4287 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 +∞cpnf 11193 ℝ*cxr 11195 < clt 11196 ≤ cle 11197 2c2 12215 ℕ0cn0 12420 ℤcz 12506 ♯chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: tgifscgr 27492 tgcgrxfr 27502 tgbtwnconn3 27561 legtrid 27575 hpgerlem 27749 |
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