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Theorem dfssr2 38953
Description: Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
Assertion
Ref Expression
dfssr2 S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))

Proof of Theorem dfssr2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epel 5528 . . . . . . 7 (𝑧 E 𝑥𝑧𝑥)
2 brvdif 38640 . . . . . . . 8 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦)
3 epel 5528 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
42, 3xchbinx 335 . . . . . . 7 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧𝑦)
51, 4anbi12i 634 . . . . . 6 ((𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑦))
65exbii 1855 . . . . 5 (∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
76notbii 321 . . . 4 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
8 dfss6 3912 . . . 4 (𝑥𝑦 ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
97, 8bitr4i 279 . . 3 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ 𝑥𝑦)
109opabbii 5146 . 2 {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
11 rnxrn 38795 . . . 4 ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1211difeq2i 4061 . . 3 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)})
13 vvdifopab 38639 . . 3 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1412, 13eqtri 2763 . 2 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
15 df-ssr 38952 . 2 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1610, 14, 153eqtr4ri 2774 1 S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1547  wex 1786  wcel 2119  Vcvv 3432  cdif 3887  wss 3890   class class class wbr 5079  {copab 5141   E cep 5524   × cxp 5623  ran crn 5626  cxrn 38548   S cssr 38560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7938  df-2nd 7939  df-ec 8642  df-xrn 38754  df-ssr 38952
This theorem is referenced by: (None)
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