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Theorem griedg0ssusgr 28255
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 π‘ˆ = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…}
Assertion
Ref Expression
griedg0ssusgr π‘ˆ βŠ† USGraph
Distinct variable group:   𝑣,𝑒
Allowed substitution hints:   π‘ˆ(𝑣,𝑒)

Proof of Theorem griedg0ssusgr
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 griedg0prc.u . . . . 5 π‘ˆ = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…}
21eleq2i 2826 . . . 4 (𝑔 ∈ π‘ˆ ↔ 𝑔 ∈ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…})
3 elopab 5485 . . . 4 (𝑔 ∈ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} ↔ βˆƒπ‘£βˆƒπ‘’(𝑔 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…))
42, 3bitri 275 . . 3 (𝑔 ∈ π‘ˆ ↔ βˆƒπ‘£βˆƒπ‘’(𝑔 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…))
5 opex 5422 . . . . . . . 8 βŸ¨π‘£, π‘’βŸ© ∈ V
65a1i 11 . . . . . . 7 (𝑒:βˆ…βŸΆβˆ… β†’ βŸ¨π‘£, π‘’βŸ© ∈ V)
7 vex 3448 . . . . . . . . 9 𝑣 ∈ V
8 vex 3448 . . . . . . . . 9 𝑒 ∈ V
97, 8opiedgfvi 28003 . . . . . . . 8 (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = 𝑒
10 f0bi 6726 . . . . . . . . 9 (𝑒:βˆ…βŸΆβˆ… ↔ 𝑒 = βˆ…)
1110biimpi 215 . . . . . . . 8 (𝑒:βˆ…βŸΆβˆ… β†’ 𝑒 = βˆ…)
129, 11eqtrid 2785 . . . . . . 7 (𝑒:βˆ…βŸΆβˆ… β†’ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…)
136, 12usgr0e 28226 . . . . . 6 (𝑒:βˆ…βŸΆβˆ… β†’ βŸ¨π‘£, π‘’βŸ© ∈ USGraph)
1413adantl 483 . . . . 5 ((𝑔 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ βŸ¨π‘£, π‘’βŸ© ∈ USGraph)
15 eleq1 2822 . . . . . 6 (𝑔 = βŸ¨π‘£, π‘’βŸ© β†’ (𝑔 ∈ USGraph ↔ βŸ¨π‘£, π‘’βŸ© ∈ USGraph))
1615adantr 482 . . . . 5 ((𝑔 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ (𝑔 ∈ USGraph ↔ βŸ¨π‘£, π‘’βŸ© ∈ USGraph))
1714, 16mpbird 257 . . . 4 ((𝑔 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ 𝑔 ∈ USGraph)
1817exlimivv 1936 . . 3 (βˆƒπ‘£βˆƒπ‘’(𝑔 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ 𝑔 ∈ USGraph)
194, 18sylbi 216 . 2 (𝑔 ∈ π‘ˆ β†’ 𝑔 ∈ USGraph)
2019ssriv 3949 1 π‘ˆ βŠ† USGraph
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283  βŸ¨cop 4593  {copab 5168  βŸΆwf 6493  β€˜cfv 6497  iEdgciedg 27990  USGraphcusgr 28142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fv 6505  df-2nd 7923  df-iedg 27992  df-usgr 28144
This theorem is referenced by:  usgrprc  28256
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