Theorem dfssr2 39209

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 5566	. . . . . . 7 ⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥)
2		brvdif 38896	. . . . . . . 8 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦)
3		epel 5566	. . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
4	2, 3	xchbinx 337	. . . . . . 7 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 ∈ 𝑦)
5	1, 4	anbi12i 639	. . . . . 6 ⊢ ((𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
6	5	exbii 1878	. . . . 5 ⊢ (∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
7	6	notbii 323	. . . 4 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
8		dfss6 3928	. . . 4 ⊢ (𝑥 ⊆ 𝑦 ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
9	7, 8	bitr4i 281	. . 3 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ 𝑥 ⊆ 𝑦)
10	9	opabbii 5179	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
11		rnxrn 39051	. . . 4 ⊢ ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}
12	11	difeq2i 4079	. . 3 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)})
13		vvdifopab 38895	. . 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 2786	. 2 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}
15		df-ssr 39208	. 2 ⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
16	10, 14, 15	3eqtr4ri 2797	1 ⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906   class class class wbr 5110  {copab 5174   E cep 5562   × cxp 5661  ran crn 5664   ⋉ cxrn 38804   S cssr 38816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 7987  df-2nd 7988  df-ec 8697  df-xrn 39010  df-ssr 39208

This theorem is referenced by: (None)