Theorem upgr1een 16106

Description: A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16103 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 7064	. . 3 ⊢ (𝐸 ≈ 2o → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
3	1, 2	syl 14	. 2 ⊢ (𝜑 → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
4		eqid 2232	. . . . 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 3679	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
9	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 ⊆ 𝑉)
10		vex 2815	. . . . . . . . 9 ⊢ 𝑢 ∈ V
11	10	prid1 3796	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢, 𝑣}
12		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 = {𝑢, 𝑣})
13	11, 12	eleqtrrid 2322	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝐸)
14	9, 13	sseldd 3238	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝑉)
15		upgr1een.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑌)
16		opexg 4343	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → ⟨𝐾, 𝐸⟩ ∈ V)
17	5, 7, 16	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐾, 𝐸⟩ ∈ V)
18		snexg 4296	. . . . . . . . 9 ⊢ (⟨𝐾, 𝐸⟩ ∈ V → {⟨𝐾, 𝐸⟩} ∈ V)
19	17, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩} ∈ V)
20		opvtxfv 16004	. . . . . . . 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 2312	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
24		vex 2815	. . . . . . . . 9 ⊢ 𝑣 ∈ V
25	24	prid2 3797	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑢, 𝑣}
26	25, 12	eleqtrrid 2322	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝐸)
27	9, 26	sseldd 3238	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝑉)
28	27, 22	eleqtrrd 2312	. . . . 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 4132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {𝑢, 𝑣} ≈ 2o)
31		pr2ne 7488	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣))
32	31	el2v 2818	. . . . . . . 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 2423	. . . . . 6 ⊢ (DECID 𝑢 = 𝑣 ↔ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
36	34, 35	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → DECID 𝑢 = 𝑣)
37		opiedgfv 16007	. . . . . . . 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 3889	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → ⟨𝐾, 𝐸⟩ = ⟨𝐾, {𝑢, 𝑣}⟩)
41	40	sneqd 3701	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {⟨𝐾, 𝐸⟩} = {⟨𝐾, {𝑢, 𝑣}⟩})
42	39, 41	eqtrd 2265	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, {𝑢, 𝑣}⟩})
43	4, 6, 23, 28, 36, 42	upgr1edc 16103	. . . 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 1947	. 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 2203   ≠ wne 2412  Vcvv 2812   ⊆ wss 3210  𝒫 cpw 3668  {csn 3688  {cpr 3689  ⟨cop 3691   class class class wbr 4108  ‘cfv 5351  2_oc2o 6640   ≈ cen 6972  Vtxcvtx 15994  iEdgciedg 15995  UPGraphcupgr 16073

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-upgren 16075

This theorem is referenced by:  umgr1een  16107  p1evtxdeqfilem  16293  p1evtxdeqfi  16294