Theorem upgrpredgv 15820

Description: An edge of a pseudograph always connects two vertices if the edge contains two sets. The two vertices/sets need not necessarily be different (loops are allowed). (Contributed by AV, 18-Nov-2021.)

Hypotheses

Ref	Expression
upgredg.v	⊢ 𝑉 = (Vtx‘𝐺)
upgredg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
upgrpredgv	⊢ ((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))

Proof of Theorem upgrpredgv

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		upgredg.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	upgredg 15818	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 {𝑀, 𝑁} = {𝑚, 𝑛})
4	3	3adant2 1019	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 {𝑀, 𝑁} = {𝑚, 𝑛})
5		preq12bg 3822	. . . . 5 ⊢ (((𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ({𝑀, 𝑁} = {𝑚, 𝑛} ↔ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) ∨ (𝑀 = 𝑛 ∧ 𝑁 = 𝑚))))
6	5	3ad2antl2 1163	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ({𝑀, 𝑁} = {𝑚, 𝑛} ↔ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) ∨ (𝑀 = 𝑛 ∧ 𝑁 = 𝑚))))
7		eleq1 2269	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚 ∈ 𝑉 ↔ 𝑀 ∈ 𝑉))
8	7	eqcoms 2209	. . . . . . . . 9 ⊢ (𝑀 = 𝑚 → (𝑚 ∈ 𝑉 ↔ 𝑀 ∈ 𝑉))
9	8	biimpd 144	. . . . . . . 8 ⊢ (𝑀 = 𝑚 → (𝑚 ∈ 𝑉 → 𝑀 ∈ 𝑉))
10		eleq1 2269	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ 𝑉 ↔ 𝑁 ∈ 𝑉))
11	10	eqcoms 2209	. . . . . . . . 9 ⊢ (𝑁 = 𝑛 → (𝑛 ∈ 𝑉 ↔ 𝑁 ∈ 𝑉))
12	11	biimpd 144	. . . . . . . 8 ⊢ (𝑁 = 𝑛 → (𝑛 ∈ 𝑉 → 𝑁 ∈ 𝑉))
13	9, 12	im2anan9 598	. . . . . . 7 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
14	13	com12 30	. . . . . 6 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
15		eleq1 2269	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (𝑛 ∈ 𝑉 ↔ 𝑀 ∈ 𝑉))
16	15	eqcoms 2209	. . . . . . . . . 10 ⊢ (𝑀 = 𝑛 → (𝑛 ∈ 𝑉 ↔ 𝑀 ∈ 𝑉))
17	16	biimpd 144	. . . . . . . . 9 ⊢ (𝑀 = 𝑛 → (𝑛 ∈ 𝑉 → 𝑀 ∈ 𝑉))
18		eleq1 2269	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑚 ∈ 𝑉 ↔ 𝑁 ∈ 𝑉))
19	18	eqcoms 2209	. . . . . . . . . 10 ⊢ (𝑁 = 𝑚 → (𝑚 ∈ 𝑉 ↔ 𝑁 ∈ 𝑉))
20	19	biimpd 144	. . . . . . . . 9 ⊢ (𝑁 = 𝑚 → (𝑚 ∈ 𝑉 → 𝑁 ∈ 𝑉))
21	17, 20	im2anan9 598	. . . . . . . 8 ⊢ ((𝑀 = 𝑛 ∧ 𝑁 = 𝑚) → ((𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
22	21	com12 30	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) → ((𝑀 = 𝑛 ∧ 𝑁 = 𝑚) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
23	22	ancoms 268	. . . . . 6 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑀 = 𝑛 ∧ 𝑁 = 𝑚) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
24	14, 23	jaod 719	. . . . 5 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) ∨ (𝑀 = 𝑛 ∧ 𝑁 = 𝑚)) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
25	24	adantl 277	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) ∨ (𝑀 = 𝑛 ∧ 𝑁 = 𝑚)) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
26	6, 25	sylbid 150	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ({𝑀, 𝑁} = {𝑚, 𝑛} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
27	26	rexlimdvva 2632	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) → (∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 {𝑀, 𝑁} = {𝑚, 𝑛} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
28	4, 27	mpd 13	1 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∃wrex 2486  {cpr 3639  ‘cfv 5285  Vtxcvtx 15696  Edgcedg 15739  UPGraphcupgr 15772

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-1o 6520  df-2o 6521  df-en 6846  df-sub 8275  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-5 9128  df-6 9129  df-7 9130  df-8 9131  df-9 9132  df-n0 9326  df-dec 9535  df-ndx 12920  df-slot 12921  df-base 12923  df-edgf 15689  df-vtx 15698  df-iedg 15699  df-edg 15740  df-upgren 15774

This theorem is referenced by: (None)