Theorem upgr1een 15978

Description: A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 15975 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 6998	. . 3 ⊢ (𝐸 ≈ 2o → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
3	1, 2	syl 14	. 2 ⊢ (𝜑 → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
4		eqid 2231	. . . . 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 3663	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
9	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 ⊆ 𝑉)
10		vex 2805	. . . . . . . . 9 ⊢ 𝑢 ∈ V
11	10	prid1 3777	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢, 𝑣}
12		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 = {𝑢, 𝑣})
13	11, 12	eleqtrrid 2321	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝐸)
14	9, 13	sseldd 3228	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝑉)
15		upgr1een.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑌)
16		opexg 4320	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → ⟨𝐾, 𝐸⟩ ∈ V)
17	5, 7, 16	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐾, 𝐸⟩ ∈ V)
18		snexg 4274	. . . . . . . . 9 ⊢ (⟨𝐾, 𝐸⟩ ∈ V → {⟨𝐾, 𝐸⟩} ∈ V)
19	17, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩} ∈ V)
20		opvtxfv 15876	. . . . . . . 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 2311	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
24		vex 2805	. . . . . . . . 9 ⊢ 𝑣 ∈ V
25	24	prid2 3778	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑢, 𝑣}
26	25, 12	eleqtrrid 2321	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝐸)
27	9, 26	sseldd 3228	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝑉)
28	27, 22	eleqtrrd 2311	. . . . 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 4112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {𝑢, 𝑣} ≈ 2o)
31		pr2ne 7397	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣))
32	31	el2v 2808	. . . . . . . 8 ⊢ ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣)
33	30, 32	sylib 122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ≠ 𝑣)
34	33	olcd 741	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
35		dcne 2413	. . . . . 6 ⊢ (DECID 𝑢 = 𝑣 ↔ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
36	34, 35	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → DECID 𝑢 = 𝑣)
37		opiedgfv 15879	. . . . . . . 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 3869	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → ⟨𝐾, 𝐸⟩ = ⟨𝐾, {𝑢, 𝑣}⟩)
41	40	sneqd 3682	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {⟨𝐾, 𝐸⟩} = {⟨𝐾, {𝑢, 𝑣}⟩})
42	39, 41	eqtrd 2264	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, {𝑢, 𝑣}⟩})
43	4, 6, 23, 28, 36, 42	upgr1edc 15975	. . . 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 1946	. 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 715  DECID wdc 841   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ≠ wne 2402  Vcvv 2802   ⊆ wss 3200  𝒫 cpw 3652  {csn 3669  {cpr 3670  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  2_oc2o 6576   ≈ cen 6907  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-upgren 15947

This theorem is referenced by:  umgr1een  15979  p1evtxdeqfilem  16165  p1evtxdeqfi  16166