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Mirrors > Home > MPE Home > Th. List > usgr2v1e2w | Structured version Visualization version GIF version |
Description: A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.) |
Ref | Expression |
---|---|
usgr2v1e2w | ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → 〈{𝐴, 𝐵}, 〈“{𝐴, 𝐵}”〉〉 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5323 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
2 | s1val 13940 | . . . 4 ⊢ ({𝐴, 𝐵} ∈ V → 〈“{𝐴, 𝐵}”〉 = {〈0, {𝐴, 𝐵}〉}) | |
3 | 1, 2 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → 〈“{𝐴, 𝐵}”〉 = {〈0, {𝐴, 𝐵}〉}) |
4 | 3 | opeq2d 4802 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → 〈{𝐴, 𝐵}, 〈“{𝐴, 𝐵}”〉〉 = 〈{𝐴, 𝐵}, {〈0, {𝐴, 𝐵}〉}〉) |
5 | prid1g 4688 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴, 𝐵}) | |
6 | prid2g 4689 | . . . . 5 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐴, 𝐵}) | |
7 | 5, 6 | anim12i 612 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) |
8 | c0ex 10623 | . . . . 5 ⊢ 0 ∈ V | |
9 | 1, 8 | pm3.2i 471 | . . . 4 ⊢ ({𝐴, 𝐵} ∈ V ∧ 0 ∈ V) |
10 | 7, 9 | jctil 520 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (({𝐴, 𝐵} ∈ V ∧ 0 ∈ V) ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵}))) |
11 | usgr1eop 26959 | . . . 4 ⊢ ((({𝐴, 𝐵} ∈ V ∧ 0 ∈ V) ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → (𝐴 ≠ 𝐵 → 〈{𝐴, 𝐵}, {〈0, {𝐴, 𝐵}〉}〉 ∈ USGraph)) | |
12 | 11 | imp 407 | . . 3 ⊢ (((({𝐴, 𝐵} ∈ V ∧ 0 ∈ V) ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) ∧ 𝐴 ≠ 𝐵) → 〈{𝐴, 𝐵}, {〈0, {𝐴, 𝐵}〉}〉 ∈ USGraph) |
13 | 10, 12 | stoic3 1768 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → 〈{𝐴, 𝐵}, {〈0, {𝐴, 𝐵}〉}〉 ∈ USGraph) |
14 | 4, 13 | eqeltrd 2910 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → 〈{𝐴, 𝐵}, 〈“{𝐴, 𝐵}”〉〉 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 {csn 4557 {cpr 4559 〈cop 4563 0cc0 10525 〈“cs1 13937 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-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 df-s1 13938 df-vtx 26710 df-iedg 26711 df-edg 26760 df-uspgr 26862 df-usgr 26863 |
This theorem is referenced by: (None) |
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