Theorem dfssr2 35733

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.

Step	Hyp	Ref	Expression
1		epel 5464	. . . . . . 7 ⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥)
2		brvdif 35516	. . . . . . . 8 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦)
3		epel 5464	. . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
4	2, 3	xchbinx 336	. . . . . . 7 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 ∈ 𝑦)
5	1, 4	anbi12i 628	. . . . . 6 ⊢ ((𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
6	5	exbii 1844	. . . . 5 ⊢ (∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
7	6	notbii 322	. . . 4 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
8		dfss6 3957	. . . 4 ⊢ (𝑥 ⊆ 𝑦 ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
9	7, 8	bitr4i 280	. . 3 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ 𝑥 ⊆ 𝑦)
10	9	opabbii 5126	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
11		rnxrn 35640	. . . 4 ⊢ ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}
12	11	difeq2i 4096	. . 3 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)})
13		vvdifopab 35515	. . 3 ⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}
14	12, 13	eqtri 2844	. 2 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}
15		df-ssr 35732	. 2 ⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
16	10, 14, 15	3eqtr4ri 2855	1 ⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3495   ∖ cdif 3933   ⊆ wss 3936   class class class wbr 5059  {copab 5121   E cep 5459   × cxp 5548  ran crn 5551   ⋉ cxrn 35446   S cssr 35450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-eprel 5460  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fo 6356  df-fv 6358  df-1st 7683  df-2nd 7684  df-ec 8285  df-xrn 35617  df-ssr 35732

This theorem is referenced by: (None)