Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for vtxdginducedm1 27325. (Contributed by AV, 17-Dec-2021.) |
Ref | Expression |
---|---|
vtxdginducedm1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdginducedm1.e | ⊢ 𝐸 = (iEdg‘𝐺) |
vtxdginducedm1.k | ⊢ 𝐾 = (𝑉 ∖ {𝑁}) |
vtxdginducedm1.i | ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} |
vtxdginducedm1.p | ⊢ 𝑃 = (𝐸 ↾ 𝐼) |
vtxdginducedm1.s | ⊢ 𝑆 = 〈𝐾, 𝑃〉 |
vtxdginducedm1.j | ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} |
Ref | Expression |
---|---|
vtxdginducedm1lem4 | ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}}) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝐸‘𝑖) = (𝐸‘𝑘)) | |
2 | 1 | eleq2d 2898 | . . . . . . 7 ⊢ (𝑖 = 𝑘 → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ (𝐸‘𝑘))) |
3 | vtxdginducedm1.j | . . . . . . 7 ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} | |
4 | 2, 3 | elrab2 3683 | . . . . . 6 ⊢ (𝑘 ∈ 𝐽 ↔ (𝑘 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑘))) |
5 | eldifsn 4719 | . . . . . . . 8 ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑊 ∈ 𝑉 ∧ 𝑊 ≠ 𝑁)) | |
6 | df-ne 3017 | . . . . . . . . 9 ⊢ (𝑊 ≠ 𝑁 ↔ ¬ 𝑊 = 𝑁) | |
7 | eleq2 2901 | . . . . . . . . . . . 12 ⊢ ((𝐸‘𝑘) = {𝑊} → (𝑁 ∈ (𝐸‘𝑘) ↔ 𝑁 ∈ {𝑊})) | |
8 | elsni 4584 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ {𝑊} → 𝑁 = 𝑊) | |
9 | 8 | eqcomd 2827 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ {𝑊} → 𝑊 = 𝑁) |
10 | 7, 9 | syl6bi 255 | . . . . . . . . . . 11 ⊢ ((𝐸‘𝑘) = {𝑊} → (𝑁 ∈ (𝐸‘𝑘) → 𝑊 = 𝑁)) |
11 | 10 | com12 32 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (𝐸‘𝑘) → ((𝐸‘𝑘) = {𝑊} → 𝑊 = 𝑁)) |
12 | 11 | con3rr3 158 | . . . . . . . . 9 ⊢ (¬ 𝑊 = 𝑁 → (𝑁 ∈ (𝐸‘𝑘) → ¬ (𝐸‘𝑘) = {𝑊})) |
13 | 6, 12 | sylbi 219 | . . . . . . . 8 ⊢ (𝑊 ≠ 𝑁 → (𝑁 ∈ (𝐸‘𝑘) → ¬ (𝐸‘𝑘) = {𝑊})) |
14 | 5, 13 | simplbiim 507 | . . . . . . 7 ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → (𝑁 ∈ (𝐸‘𝑘) → ¬ (𝐸‘𝑘) = {𝑊})) |
15 | 14 | com12 32 | . . . . . 6 ⊢ (𝑁 ∈ (𝐸‘𝑘) → (𝑊 ∈ (𝑉 ∖ {𝑁}) → ¬ (𝐸‘𝑘) = {𝑊})) |
16 | 4, 15 | simplbiim 507 | . . . . 5 ⊢ (𝑘 ∈ 𝐽 → (𝑊 ∈ (𝑉 ∖ {𝑁}) → ¬ (𝐸‘𝑘) = {𝑊})) |
17 | 16 | impcom 410 | . . . 4 ⊢ ((𝑊 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑘 ∈ 𝐽) → ¬ (𝐸‘𝑘) = {𝑊}) |
18 | 17 | ralrimiva 3182 | . . 3 ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → ∀𝑘 ∈ 𝐽 ¬ (𝐸‘𝑘) = {𝑊}) |
19 | rabeq0 4338 | . . 3 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}} = ∅ ↔ ∀𝑘 ∈ 𝐽 ¬ (𝐸‘𝑘) = {𝑊}) | |
20 | 18, 19 | sylibr 236 | . 2 ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}} = ∅) |
21 | vtxdginducedm1.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
22 | 21 | fvexi 6684 | . . . . 5 ⊢ 𝐸 ∈ V |
23 | 22 | dmex 7616 | . . . 4 ⊢ dom 𝐸 ∈ V |
24 | 3, 23 | rab2ex 5238 | . . 3 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}} ∈ V |
25 | hasheq0 13725 | . . 3 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}} ∈ V → ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}}) = 0 ↔ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}} = ∅)) | |
26 | 24, 25 | ax-mp 5 | . 2 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}}) = 0 ↔ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}} = ∅) |
27 | 20, 26 | sylibr 236 | 1 ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}}) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∉ wnel 3123 ∀wral 3138 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∅c0 4291 {csn 4567 〈cop 4573 dom cdm 5555 ↾ cres 5557 ‘cfv 6355 0cc0 10537 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 |
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-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: vtxdginducedm1 27325 |
Copyright terms: Public domain | W3C validator |