Theorem spr0el 48231

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 48225	. 2 ⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
2		sprssspr 48230	. . . . 5 ⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
3	2	sseli 3933	. . . 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 3065	. . 3 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
6		df-nel 3065	. . 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 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741   ∉ wnel 3064  ∅c0 4286  {cpr 4591  ‘cfv 6536  Pairscspr 48226

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-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

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-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-spr 48227

This theorem is referenced by: (None)