Theorem griedg0ssusgr 29350

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 2829	. . . 4 ⊢ (𝑔 ∈ 𝑈 ↔ 𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅})
3		elopab 5483	. . . 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 5419	. . . . . . . 8 ⊢ ⟨𝑣, 𝑒⟩ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ V)
7		vex 3446	. . . . . . . . 9 ⊢ 𝑣 ∈ V
8		vex 3446	. . . . . . . . 9 ⊢ 𝑒 ∈ V
9	7, 8	opiedgfvi 29095	. . . . . . . 8 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10		f0bi 6725	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
11	10	biimpi 216	. . . . . . . 8 ⊢ (𝑒:∅⟶∅ → 𝑒 = ∅)
12	9, 11	eqtrid 2784	. . . . . . 7 ⊢ (𝑒:∅⟶∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
13	6, 12	usgr0e 29321	. . . . . 6 ⊢ (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
14	13	adantl 481	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
15		eleq1 2825	. . . . . 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 1934	. . 3 ⊢ (∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph)
19	4, 18	sylbi 217	. 2 ⊢ (𝑔 ∈ 𝑈 → 𝑔 ∈ USGraph)
20	19	ssriv 3939	1 ⊢ 𝑈 ⊆ USGraph

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ⟨cop 4588  {copab 5162  ⟶wf 6496  ‘cfv 6500  iEdgciedg 29082  USGraphcusgr 29234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508  df-2nd 7944  df-iedg 29084  df-usgr 29236

This theorem is referenced by:  usgrprc  29351