Theorem upgr1een 16348

Description: A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16345 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 7106	. . 3 ⊢ (𝐸 ≈ 2o → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
3	1, 2	syl 14	. 2 ⊢ (𝜑 → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
4		eqid 2238	. . . . 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 3699	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
9	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 ⊆ 𝑉)
10		vex 2824	. . . . . . . . 9 ⊢ 𝑢 ∈ V
11	10	prid1 3816	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢, 𝑣}
12		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 = {𝑢, 𝑣})
13	11, 12	eleqtrrid 2328	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝐸)
14	9, 13	sseldd 3249	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝑉)
15		upgr1een.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑌)
16		opexg 4366	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → ⟨𝐾, 𝐸⟩ ∈ V)
17	5, 7, 16	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐾, 𝐸⟩ ∈ V)
18		snexg 4319	. . . . . . . . 9 ⊢ (⟨𝐾, 𝐸⟩ ∈ V → {⟨𝐾, 𝐸⟩} ∈ V)
19	17, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩} ∈ V)
20		opvtxfv 16246	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
21	15, 19, 20	syl2anc 415	. . . . . . 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 2318	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
24		vex 2824	. . . . . . . . 9 ⊢ 𝑣 ∈ V
25	24	prid2 3817	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑢, 𝑣}
26	25, 12	eleqtrrid 2328	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝐸)
27	9, 26	sseldd 3249	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝑉)
28	27, 22	eleqtrrd 2318	. . . . 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 4152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {𝑢, 𝑣} ≈ 2o)
31		pr2ne 7532	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣))
32	31	el2v 2827	. . . . . . . 8 ⊢ ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣)
33	30, 32	sylib 122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ≠ 𝑣)
34	33	olcd 746	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
35		dcne 2431	. . . . . 6 ⊢ (DECID 𝑢 = 𝑣 ↔ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
36	34, 35	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → DECID 𝑢 = 𝑣)
37		opiedgfv 16249	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
38	15, 19, 37	syl2anc 415	. . . . . . 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 3909	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → ⟨𝐾, 𝐸⟩ = ⟨𝐾, {𝑢, 𝑣}⟩)
41	40	sneqd 3721	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {⟨𝐾, 𝐸⟩} = {⟨𝐾, {𝑢, 𝑣}⟩})
42	39, 41	eqtrd 2271	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, {𝑢, 𝑣}⟩})
43	4, 6, 23, 28, 36, 42	upgr1edc 16345	. . . 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 1953	. 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 720  DECID wdc 846   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688  {csn 3708  {cpr 3709  ⟨cop 3711   class class class wbr 4128  ‘cfv 5375  2_oc2o 6675   ≈ cen 7014  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-1o 6681  df-2o 6682  df-er 6801  df-en 7017  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-upgren 16317

This theorem is referenced by:  umgr1een  16349  p1evtxdeqfilem  16535  p1evtxdeqfi  16536