Theorem griedg0ssusgr 40489

Description: The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.)

Hypothesis

Ref	Expression
griedg0prc.u	⊢ 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}

Assertion

Ref	Expression
griedg0ssusgr	⊢ 𝑈 ⊆ USGraph

Distinct variable group:   𝑣,𝑒

Allowed substitution hints:   𝑈(𝑣,𝑒)

Proof of Theorem griedg0ssusgr

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		griedg0prc.u	. . . . 5 ⊢ 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
2	1	eleq2i 2680	. . . 4 ⊢ (𝑔 ∈ 𝑈 ↔ 𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅})
3		elopab 4908	. . . 4 ⊢ (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
4	2, 3	bitri 263	. . 3 ⊢ (𝑔 ∈ 𝑈 ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
5		opex 4859	. . . . . . . 8 ⊢ ⟨𝑣, 𝑒⟩ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ V)
7		vex 3176	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8		vex 3176	. . . . . . . . 9 ⊢ 𝑒 ∈ V
9		opiedgfv 25684	. . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒)
10	7, 8, 9	mp2an 704	. . . . . . . 8 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
11		f0bi 6001	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
12	11	biimpi 205	. . . . . . . 8 ⊢ (𝑒:∅⟶∅ → 𝑒 = ∅)
13	10, 12	syl5eq 2656	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
14	6, 13	usgr0e 40462	. . . . . 6 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph )
15	14	adantl 481	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ⟨𝑣, 𝑒⟩ ∈ USGraph )
16		eleq1 2676	. . . . . 6 ⊢ (𝑔 = ⟨𝑣, 𝑒⟩ → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
17	16	adantr 480	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
18	15, 17	mpbird 246	. . . 4 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph )
19	18	exlimivv 1847	. . 3 ⊢ (∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph )
20	4, 19	sylbi 206	. 2 ⊢ (𝑔 ∈ 𝑈 → 𝑔 ∈ USGraph )
21	20	ssriv 3572	1 ⊢ 𝑈 ⊆ USGraph

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131  {copab 4642  ⟶wf 5800  ‘cfv 5804  iEdgciedg 25674   USGraph cusgr 40379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-2nd 7060  df-iedg 25676  df-usgr 40381

This theorem is referenced by:  usgrprc  40490