Theorem griedg0ssusgr 29876

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 2856	. . . 4 ⊢ (𝑔 ∈ 𝑈 ↔ 𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅})
3		elopab 5505	. . . 4 ⊢ (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
4	2, 3	bitri 278	. . 3 ⊢ (𝑔 ∈ 𝑈 ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
5		opex 5436	. . . . . . . 8 ⊢ ⟨𝑣, 𝑒⟩ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ V)
7		vex 3458	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8		vex 3458	. . . . . . . . 9 ⊢ 𝑒 ∈ V
9	7, 8	opiedgfvi 29618	. . . . . . . 8 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10		f0bi 6787	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
11	10	biimpi 219	. . . . . . . 8 ⊢ (𝑒:∅⟶∅ → 𝑒 = ∅)
12	9, 11	eqtrid 2810	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
13	6, 12	usgr0e 29847	. . . . . 6 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
14	13	adantl 487	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
15		eleq1 2852	. . . . . 6 ⊢ (𝑔 = ⟨𝑣, 𝑒⟩ → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
16	15	adantr 486	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
17	14, 16	mpbird 260	. . . 4 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph)
18	17	exlimivv 1967	. . 3 ⊢ (∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph)
19	4, 18	sylbi 220	. 2 ⊢ (𝑔 ∈ 𝑈 → 𝑔 ∈ USGraph)
20	19	ssriv 3917	1 ⊢ 𝑈 ⊆ USGraph

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ⟨cop 4838  {copab 5170  ⟶wf 6554  ‘cfv 6558  iEdgciedg 29605  USGraphcusgr 29760

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fv 6566  df-2nd 8026  df-iedg 29607  df-usgr 29762

This theorem is used by:  usgrprc  29877