Theorem griedg0ssusgr 29199

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 2821	. . . 4 ⊢ (𝑔 ∈ 𝑈 ↔ 𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅})
3		elopab 5490	. . . 4 ⊢ (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
4	2, 3	bitri 275	. . 3 ⊢ (𝑔 ∈ 𝑈 ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
5		opex 5427	. . . . . . . 8 ⊢ ⟨𝑣, 𝑒⟩ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ V)
7		vex 3454	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8		vex 3454	. . . . . . . . 9 ⊢ 𝑒 ∈ V
9	7, 8	opiedgfvi 28944	. . . . . . . 8 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10		f0bi 6746	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
11	10	biimpi 216	. . . . . . . 8 ⊢ (𝑒:∅⟶∅ → 𝑒 = ∅)
12	9, 11	eqtrid 2777	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
13	6, 12	usgr0e 29170	. . . . . 6 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
14	13	adantl 481	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
15		eleq1 2817	. . . . . 6 ⊢ (𝑔 = ⟨𝑣, 𝑒⟩ → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
16	15	adantr 480	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
17	14, 16	mpbird 257	. . . 4 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph)
18	17	exlimivv 1932	. . 3 ⊢ (∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph)
19	4, 18	sylbi 217	. 2 ⊢ (𝑔 ∈ 𝑈 → 𝑔 ∈ USGraph)
20	19	ssriv 3953	1 ⊢ 𝑈 ⊆ USGraph

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  ⟨cop 4598  {copab 5172  ⟶wf 6510  ‘cfv 6514  iEdgciedg 28931  USGraphcusgr 29083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fv 6522  df-2nd 7972  df-iedg 28933  df-usgr 29085

This theorem is referenced by:  usgrprc  29200