![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > upgredgpr | Structured version Visualization version GIF version |
Description: If a proper pair (of vertices) is a subset of an edge in a pseudograph, the pair is the edge. (Contributed by AV, 30-Dec-2020.) |
Ref | Expression |
---|---|
upgredg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgredg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
upgredgpr | ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → {𝐴, 𝐵} = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgredg.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | upgredg.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | upgredg 26930 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏}) |
4 | 3 | 3adant3 1129 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏}) |
5 | ssprsseq 4718 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝑎, 𝑏} ↔ {𝐴, 𝐵} = {𝑎, 𝑏})) | |
6 | 5 | biimpd 232 | . . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏})) |
7 | sseq2 3941 | . . . . . . . . . 10 ⊢ (𝐶 = {𝑎, 𝑏} → ({𝐴, 𝐵} ⊆ 𝐶 ↔ {𝐴, 𝐵} ⊆ {𝑎, 𝑏})) | |
8 | eqeq2 2810 | . . . . . . . . . 10 ⊢ (𝐶 = {𝑎, 𝑏} → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴, 𝐵} = {𝑎, 𝑏})) | |
9 | 7, 8 | imbi12d 348 | . . . . . . . . 9 ⊢ (𝐶 = {𝑎, 𝑏} → (({𝐴, 𝐵} ⊆ 𝐶 → {𝐴, 𝐵} = 𝐶) ↔ ({𝐴, 𝐵} ⊆ {𝑎, 𝑏} → {𝐴, 𝐵} = {𝑎, 𝑏}))) |
10 | 6, 9 | syl5ibr 249 | . . . . . . . 8 ⊢ (𝐶 = {𝑎, 𝑏} → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ 𝐶 → {𝐴, 𝐵} = 𝐶))) |
11 | 10 | com23 86 | . . . . . . 7 ⊢ (𝐶 = {𝑎, 𝑏} → ({𝐴, 𝐵} ⊆ 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = 𝐶))) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐶 = {𝑎, 𝑏} → ({𝐴, 𝐵} ⊆ 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = 𝐶)))) |
13 | 12 | rexlimivv 3251 | . . . . 5 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏} → ({𝐴, 𝐵} ⊆ 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = 𝐶))) |
14 | 13 | com12 32 | . . . 4 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏} → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = 𝐶))) |
15 | 14 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏} → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = 𝐶))) |
16 | 4, 15 | mpd 15 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = 𝐶)) |
17 | 16 | imp 410 | 1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → {𝐴, 𝐵} = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ⊆ wss 3881 {cpr 4527 ‘cfv 6324 Vtxcvtx 26789 Edgcedg 26840 UPGraphcupgr 26873 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-edg 26841 df-upgr 26875 |
This theorem is referenced by: nbupgr 27134 nbumgrvtx 27136 upgriswlk 27430 upgrwlkupwlk 44368 |
Copyright terms: Public domain | W3C validator |