Theorem spr0el 48115

Description: The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)

Assertion

Ref	Expression
spr0el	⊢ ∅ ∉ (Pairs‘𝑉)

Proof of Theorem spr0el

Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		spr0nelg 48109	. 2 ⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
2		sprssspr 48114	. . . . 5 ⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
3	2	sseli 3941	. . . 4 ⊢ (∅ ∈ (Pairs‘𝑉) → ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
4	3	con3i 155	. . 3 ⊢ (¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} → ¬ ∅ ∈ (Pairs‘𝑉))
5		df-nel 3071	. . 3 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
6		df-nel 3071	. . 3 ⊢ (∅ ∉ (Pairs‘𝑉) ↔ ¬ ∅ ∈ (Pairs‘𝑉))
7	4, 5, 6	3imtr4i 295	. 2 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} → ∅ ∉ (Pairs‘𝑉))
8	1, 7	ax-mp 5	1 ⊢ ∅ ∉ (Pairs‘𝑉)

Syntax hints:  ¬ wn 3   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747   ∉ wnel 3070  ∅c0 4294  {cpr 4593  ‘cfv 6534  Pairscspr 48110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-spr 48111

This theorem is referenced by: (None)