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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 6919 | . . . . 5 ⊢ 𝑃 ∈ V | 
| 3 | hashv01gt1 14385 | . . . . 5 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) | 
| 5 | 3orass 1089 | . . . 4 ⊢ (((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)))) | |
| 6 | 4, 5 | mpbi 230 | . . 3 ⊢ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) | 
| 7 | 1p1e2 12392 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
| 8 | 1z 12649 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
| 9 | nn0z 12640 | . . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℤ) | |
| 10 | zltp1le 12669 | . . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑃) ∈ ℤ) → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) | |
| 11 | 8, 9, 10 | sylancr 587 | . . . . . . . 8 ⊢ ((♯‘𝑃) ∈ ℕ0 → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) | 
| 12 | 11 | biimpac 478 | . . . . . . 7 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → (1 + 1) ≤ (♯‘𝑃)) | 
| 13 | 7, 12 | eqbrtrrid 5178 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → 2 ≤ (♯‘𝑃)) | 
| 14 | 2re 12341 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
| 15 | 14 | rexri 11320 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* | 
| 16 | pnfge 13173 | . . . . . . . . 9 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
| 17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ 2 ≤ +∞ | 
| 18 | breq2 5146 | . . . . . . . 8 ⊢ ((♯‘𝑃) = +∞ → (2 ≤ (♯‘𝑃) ↔ 2 ≤ +∞)) | |
| 19 | 17, 18 | mpbiri 258 | . . . . . . 7 ⊢ ((♯‘𝑃) = +∞ → 2 ≤ (♯‘𝑃)) | 
| 20 | 19 | adantl 481 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) = +∞) → 2 ≤ (♯‘𝑃)) | 
| 21 | hashnn0pnf 14382 | . . . . . . 7 ⊢ (𝑃 ∈ V → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) | |
| 22 | 2, 21 | mp1i 13 | . . . . . 6 ⊢ (1 < (♯‘𝑃) → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) | 
| 23 | 13, 20, 22 | mpjaodan 960 | . . . . 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 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 28 | ne0i 4340 | . . 3 ⊢ (𝐴 ∈ 𝑃 → 𝑃 ≠ ∅) | |
| 29 | hasheq0 14403 | . . . . . 6 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ↔ 𝑃 = ∅)) | |
| 30 | 2, 29 | ax-mp 5 | . . . . 5 ⊢ ((♯‘𝑃) = 0 ↔ 𝑃 = ∅) | 
| 31 | 30 | biimpi 216 | . . . 4 ⊢ ((♯‘𝑃) = 0 → 𝑃 = ∅) | 
| 32 | 31 | necon3ai 2964 | . . 3 ⊢ (𝑃 ≠ ∅ → ¬ (♯‘𝑃) = 0) | 
| 33 | biorf 936 | . . 3 ⊢ (¬ (♯‘𝑃) = 0 → (((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))))) | |
| 34 | 27, 28, 32, 33 | 4syl 19 | . 2 ⊢ (𝜑 → (((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))))) | 
| 35 | 26, 34 | mpbird 257 | 1 ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∅c0 4332 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 + caddc 11159 +∞cpnf 11293 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 2c2 12322 ℕ0cn0 12528 ℤcz 12615 ♯chash 14370 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-hash 14371 | 
| This theorem is referenced by: tgifscgr 28517 tgcgrxfr 28527 tgbtwnconn3 28586 legtrid 28600 hpgerlem 28774 | 
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