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Theorem dfssr2 38946
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 5521 . . . . . . 7 (𝑧 E 𝑥𝑧𝑥)
2 brvdif 38633 . . . . . . . 8 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦)
3 epel 5521 . . . . . . . 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 3905 . . . 4 (𝑥𝑦 ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
97, 8bitr4i 279 . . 3 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ 𝑥𝑦)
109opabbii 5139 . 2 {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
11 rnxrn 38788 . . . 4 ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1211difeq2i 4054 . . 3 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)})
13 vvdifopab 38632 . . 3 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1412, 13eqtri 2762 . 2 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
15 df-ssr 38945 . 2 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1610, 14, 153eqtr4ri 2773 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 3431  cdif 3880  wss 3883   class class class wbr 5072  {copab 5134   E cep 5517   × cxp 5616  ran crn 5619  cxrn 38541   S cssr 38553
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 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678
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 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-eprel 5518  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493  df-1st 7931  df-2nd 7932  df-ec 8635  df-xrn 38747  df-ssr 38945
This theorem is referenced by: (None)
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