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Mirrors > Home > MPE Home > Th. List > usgrnloopvALT | Structured version Visualization version GIF version |
Description: Alternate proof of usgrnloopv 26909, not using umgrnloopv 26818. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
usgrnloopv.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
usgrnloopvALT | ⊢ ((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prnzg 4705 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝑊 → {𝑀, 𝑁} ≠ ∅) | |
2 | 1 | adantl 482 | . . . . . . 7 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → {𝑀, 𝑁} ≠ ∅) |
3 | neeq1 3075 | . . . . . . . 8 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((𝐸‘𝑋) ≠ ∅ ↔ {𝑀, 𝑁} ≠ ∅)) | |
4 | 3 | adantr 481 | . . . . . . 7 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) ≠ ∅ ↔ {𝑀, 𝑁} ≠ ∅)) |
5 | 2, 4 | mpbird 258 | . . . . . 6 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐸‘𝑋) ≠ ∅) |
6 | fvfundmfvn0 6701 | . . . . . 6 ⊢ ((𝐸‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐸 ∧ Fun (𝐸 ↾ {𝑋}))) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝑋 ∈ dom 𝐸 ∧ Fun (𝐸 ↾ {𝑋}))) |
8 | usgrnloopv.e | . . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺) | |
9 | 8 | usgredg2 26901 | . . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸‘𝑋)) = 2) |
10 | fveq2 6663 | . . . . . . . . . . . 12 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → (♯‘(𝐸‘𝑋)) = (♯‘{𝑀, 𝑁})) | |
11 | 10 | eqeq1d 2820 | . . . . . . . . . . 11 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((♯‘(𝐸‘𝑋)) = 2 ↔ (♯‘{𝑀, 𝑁}) = 2)) |
12 | eqid 2818 | . . . . . . . . . . . . 13 ⊢ {𝑀, 𝑁} = {𝑀, 𝑁} | |
13 | 12 | hashprdifel 13747 | . . . . . . . . . . . 12 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁} ∧ 𝑀 ≠ 𝑁)) |
14 | 13 | simp3d 1136 | . . . . . . . . . . 11 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → 𝑀 ≠ 𝑁) |
15 | 11, 14 | syl6bi 254 | . . . . . . . . . 10 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((♯‘(𝐸‘𝑋)) = 2 → 𝑀 ≠ 𝑁)) |
16 | 15 | adantr 481 | . . . . . . . . 9 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → ((♯‘(𝐸‘𝑋)) = 2 → 𝑀 ≠ 𝑁)) |
17 | 9, 16 | syl5com 31 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → 𝑀 ≠ 𝑁)) |
18 | 17 | expcom 414 | . . . . . . 7 ⊢ (𝑋 ∈ dom 𝐸 → (𝐺 ∈ USGraph → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → 𝑀 ≠ 𝑁))) |
19 | 18 | com23 86 | . . . . . 6 ⊢ (𝑋 ∈ dom 𝐸 → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐺 ∈ USGraph → 𝑀 ≠ 𝑁))) |
20 | 19 | adantr 481 | . . . . 5 ⊢ ((𝑋 ∈ dom 𝐸 ∧ Fun (𝐸 ↾ {𝑋})) → (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐺 ∈ USGraph → 𝑀 ≠ 𝑁))) |
21 | 7, 20 | mpcom 38 | . . . 4 ⊢ (((𝐸‘𝑋) = {𝑀, 𝑁} ∧ 𝑀 ∈ 𝑊) → (𝐺 ∈ USGraph → 𝑀 ≠ 𝑁)) |
22 | 21 | ex 413 | . . 3 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑊 → (𝐺 ∈ USGraph → 𝑀 ≠ 𝑁))) |
23 | 22 | com13 88 | . 2 ⊢ (𝐺 ∈ USGraph → (𝑀 ∈ 𝑊 → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁))) |
24 | 23 | imp 407 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 {csn 4557 {cpr 4559 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 ‘cfv 6348 2c2 11680 ♯chash 13678 iEdgciedg 26709 USGraphcusgr 26861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 df-umgr 26795 df-usgr 26863 |
This theorem is referenced by: (None) |
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