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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 6684 | . . . . 5 ⊢ 𝑃 ∈ V |
3 | hashv01gt1 13706 | . . . . 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 11763 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
8 | 1z 12013 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
9 | nn0z 12006 | . . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℤ) | |
10 | zltp1le 12033 | . . . . . . . . 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 5102 | . . . . . 6 ⊢ ((1 < (♯‘𝑃) ∧ (♯‘𝑃) ∈ ℕ0) → 2 ≤ (♯‘𝑃)) |
14 | 2re 11712 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
15 | 14 | rexri 10699 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
16 | pnfge 12526 | . . . . . . . . 9 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ 2 ≤ +∞ |
18 | breq2 5070 | . . . . . . . 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 13703 | . . . . . . 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 4300 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → 𝑃 ≠ ∅) | |
29 | hasheq0 13725 | . . . . . . 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 3041 | . . . 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 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 2c2 11693 ℕ0cn0 11898 ℤcz 11982 ♯chash 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: tgifscgr 26294 tgcgrxfr 26304 tgbtwnconn3 26363 legtrid 26377 hpgerlem 26551 |
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