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Theorem dfssr2 34743
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 5228 . . . . . . 7 (𝑧 E 𝑥𝑧𝑥)
2 brvdif 34525 . . . . . . . 8 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦)
3 epel 5228 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
42, 3xchbinx 326 . . . . . . 7 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧𝑦)
51, 4anbi12i 621 . . . . . 6 ((𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑦))
65exbii 1944 . . . . 5 (∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
76notbii 312 . . . 4 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
8 dfss6 3788 . . . 4 (𝑥𝑦 ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
97, 8bitr4i 270 . . 3 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ 𝑥𝑦)
109opabbii 4910 . 2 {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
11 rnxrn 34650 . . . 4 ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1211difeq2i 3923 . . 3 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)})
13 vvdifopab 34524 . . 3 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1412, 13eqtri 2821 . 2 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
15 df-ssr 34742 . 2 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1610, 14, 153eqtr4ri 2832 1 S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 385   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  cdif 3766  wss 3769   class class class wbr 4843  {copab 4905   E cep 5224   × cxp 5310  ran crn 5313  cxrn 34468   S cssr 34472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-eprel 5225  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7401  df-2nd 7402  df-ec 7984  df-xrn 34627  df-ssr 34742
This theorem is referenced by: (None)
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