Theorem upgr1een 16136

Description: A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16133 for a different way of specifying a graph with one edge. (Contributed by Jim Kingdon, 18-Mar-2026.)

Hypotheses

Ref	Expression
upgr1een.k	⊢ (𝜑 → 𝐾 ∈ 𝑋)
upgr1een.v	⊢ (𝜑 → 𝑉 ∈ 𝑌)
upgr1een.e	⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)
upgr1een.2o	⊢ (𝜑 → 𝐸 ≈ 2o)

Assertion

Ref	Expression
upgr1een	⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)

Proof of Theorem upgr1een

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgr1een.2o	. . 3 ⊢ (𝜑 → 𝐸 ≈ 2o)
2		en2 7067	. . 3 ⊢ (𝐸 ≈ 2o → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
3	1, 2	syl 14	. 2 ⊢ (𝜑 → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
4		eqid 2234	. . . . 5 ⊢ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)
5		upgr1een.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
6	5	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐾 ∈ 𝑋)
7		upgr1een.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)
8	7	elpwid 3682	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
9	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 ⊆ 𝑉)
10		vex 2818	. . . . . . . . 9 ⊢ 𝑢 ∈ V
11	10	prid1 3799	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢, 𝑣}
12		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 = {𝑢, 𝑣})
13	11, 12	eleqtrrid 2324	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝐸)
14	9, 13	sseldd 3241	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝑉)
15		upgr1een.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑌)
16		opexg 4346	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → ⟨𝐾, 𝐸⟩ ∈ V)
17	5, 7, 16	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐾, 𝐸⟩ ∈ V)
18		snexg 4299	. . . . . . . . 9 ⊢ (⟨𝐾, 𝐸⟩ ∈ V → {⟨𝐾, 𝐸⟩} ∈ V)
19	17, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩} ∈ V)
20		opvtxfv 16034	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
21	15, 19, 20	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
22	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
23	14, 22	eleqtrrd 2314	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
24		vex 2818	. . . . . . . . 9 ⊢ 𝑣 ∈ V
25	24	prid2 3800	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑢, 𝑣}
26	25, 12	eleqtrrid 2324	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝐸)
27	9, 26	sseldd 3241	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝑉)
28	27, 22	eleqtrrd 2314	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
29	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 ≈ 2o)
30	12, 29	eqbrtrrd 4135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {𝑢, 𝑣} ≈ 2o)
31		pr2ne 7491	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣))
32	31	el2v 2821	. . . . . . . 8 ⊢ ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣)
33	30, 32	sylib 122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ≠ 𝑣)
34	33	olcd 742	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
35		dcne 2425	. . . . . 6 ⊢ (DECID 𝑢 = 𝑣 ↔ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
36	34, 35	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → DECID 𝑢 = 𝑣)
37		opiedgfv 16037	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
38	15, 19, 37	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
39	38	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
40	12	opeq2d 3892	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → ⟨𝐾, 𝐸⟩ = ⟨𝐾, {𝑢, 𝑣}⟩)
41	40	sneqd 3704	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {⟨𝐾, 𝐸⟩} = {⟨𝐾, {𝑢, 𝑣}⟩})
42	39, 41	eqtrd 2267	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, {𝑢, 𝑣}⟩})
43	4, 6, 23, 28, 36, 42	upgr1edc 16133	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)
44	43	ex 115	. . 3 ⊢ (𝜑 → (𝐸 = {𝑢, 𝑣} → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph))
45	44	exlimdvv 1949	. 2 ⊢ (𝜑 → (∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣} → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph))
46	3, 45	mpd 13	1 ⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398  ∃wex 1541   ∈ wcel 2205   ≠ wne 2414  Vcvv 2815   ⊆ wss 3213  𝒫 cpw 3671  {csn 3691  {cpr 3692  ⟨cop 3694   class class class wbr 4111  ‘cfv 5354  2_oc2o 6643   ≈ cen 6975  Vtxcvtx 16024  iEdgciedg 16025  UPGraphcupgr 16103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-upgren 16105

This theorem is referenced by:  umgr1een  16137  p1evtxdeqfilem  16323  p1evtxdeqfi  16324