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Mirrors > Home > MPE Home > Th. List > tgldim0eq | Structured version Visualization version GIF version |
Description: In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 11-Apr-2019.) |
Ref | Expression |
---|---|
tgldim0.g | ⊢ 𝑃 = (𝐸‘𝐹) |
tgldim0.p | ⊢ (𝜑 → (♯‘𝑃) = 1) |
tgldim0.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgldim0.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
tgldim0eq | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgldim0.p | . . 3 ⊢ (𝜑 → (♯‘𝑃) = 1) | |
2 | tgldim0.g | . . . . 5 ⊢ 𝑃 = (𝐸‘𝐹) | |
3 | 2 | fvexi 6853 | . . . 4 ⊢ 𝑃 ∈ V |
4 | hash1snb 14311 | . . . 4 ⊢ (𝑃 ∈ V → ((♯‘𝑃) = 1 ↔ ∃𝑥 𝑃 = {𝑥})) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ((♯‘𝑃) = 1 ↔ ∃𝑥 𝑃 = {𝑥}) |
6 | 1, 5 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑥 𝑃 = {𝑥}) |
7 | tgldim0.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝐴 ∈ 𝑃) |
9 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝑃 = {𝑥}) | |
10 | 8, 9 | eleqtrd 2840 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝐴 ∈ {𝑥}) |
11 | elsni 4601 | . . . 4 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝐴 = 𝑥) |
13 | tgldim0.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
14 | 13 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝐵 ∈ 𝑃) |
15 | 14, 9 | eleqtrd 2840 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝐵 ∈ {𝑥}) |
16 | elsni 4601 | . . . 4 ⊢ (𝐵 ∈ {𝑥} → 𝐵 = 𝑥) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝐵 = 𝑥) |
18 | 12, 17 | eqtr4d 2779 | . 2 ⊢ ((𝜑 ∧ 𝑃 = {𝑥}) → 𝐴 = 𝐵) |
19 | 6, 18 | exlimddv 1938 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3443 {csn 4584 ‘cfv 6493 1c1 11048 ♯chash 14222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-oadd 8412 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-dju 9833 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-hash 14223 |
This theorem is referenced by: tgldim0itv 27332 tgldim0cgr 27333 tglndim0 27457 |
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