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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 6687 | . . . . 5 ⊢ 𝑃 ∈ V |
3 | hashv01gt1 13708 | . . . . 5 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) |
5 | 3orass 1086 | . . . 4 ⊢ (((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) ↔ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)))) | |
6 | 4, 5 | mpbi 232 | . . 3 ⊢ ((♯‘𝑃) = 0 ∨ ((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃))) |
7 | 1p1e2 11765 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
8 | 1z 12015 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
9 | nn0z 12008 | . . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℤ) | |
10 | zltp1le 12035 | . . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑃) ∈ ℤ) → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) | |
11 | 8, 9, 10 | sylancr 589 | . . . . . . . 8 ⊢ ((♯‘𝑃) ∈ ℕ0 → (1 < (♯‘𝑃) ↔ (1 + 1) ≤ (♯‘𝑃))) |
12 | 11 | biimpac 481 | . . . . . . 7 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → (1 + 1) ≤ (♯‘𝑃)) |
13 | 7, 12 | eqbrtrrid 5105 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → 2 ≤ (♯‘𝑃)) |
14 | 2re 11714 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
15 | 14 | rexri 10702 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
16 | pnfge 12528 | . . . . . . . . 9 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ 2 ≤ +∞ |
18 | breq2 5073 | . . . . . . . 8 ⊢ ((♯‘𝑃) = +∞ → (2 ≤ (♯‘𝑃) ↔ 2 ≤ +∞)) | |
19 | 17, 18 | mpbiri 260 | . . . . . . 7 ⊢ ((♯‘𝑃) = +∞ → 2 ≤ (♯‘𝑃)) |
20 | 19 | adantl 484 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) = +∞) → 2 ≤ (♯‘𝑃)) |
21 | hashnn0pnf 13705 | . . . . . . 7 ⊢ (𝑃 ∈ V → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) | |
22 | 2, 21 | mp1i 13 | . . . . . 6 ⊢ (1 < (♯‘𝑃) → ((♯‘𝑃) ∈ ℕ0 ∨ (♯‘𝑃) = +∞)) |
23 | 13, 20, 22 | mpjaodan 955 | . . . . 5 ⊢ (1 < (♯‘𝑃) → 2 ≤ (♯‘𝑃)) |
24 | 23 | orim2i 907 | . . . 4 ⊢ (((♯‘𝑃) = 1 ∨ 1 < (♯‘𝑃)) → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
25 | 24 | orim2i 907 | . . 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 4303 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → 𝑃 ≠ ∅) | |
29 | hasheq0 13727 | . . . . . . 7 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 0 ↔ 𝑃 = ∅)) | |
30 | 2, 29 | ax-mp 5 | . . . . . 6 ⊢ ((♯‘𝑃) = 0 ↔ 𝑃 = ∅) |
31 | 30 | biimpi 218 | . . . . 5 ⊢ ((♯‘𝑃) = 0 → 𝑃 = ∅) |
32 | 31 | necon3ai 3044 | . . . 4 ⊢ (𝑃 ≠ ∅ → ¬ (♯‘𝑃) = 0) |
33 | 27, 28, 32 | 3syl 18 | . . 3 ⊢ (𝜑 → ¬ (♯‘𝑃) = 0) |
34 | biorf 933 | . . 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 259 | 1 ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∅c0 4294 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 + caddc 10543 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ♯chash 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: tgifscgr 26297 tgcgrxfr 26307 tgbtwnconn3 26366 legtrid 26380 hpgerlem 26554 |
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