Theorem spr0el 47842

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 47836	. 2 ⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
2		sprssspr 47841	. . . . 5 ⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
3	2	sseli 3931	. . . 4 ⊢ (∅ ∈ (Pairs‘𝑉) → ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
4	3	con3i 154	. . 3 ⊢ (¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} → ¬ ∅ ∈ (Pairs‘𝑉))
5		df-nel 3038	. . 3 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
6		df-nel 3038	. . 3 ⊢ (∅ ∉ (Pairs‘𝑉) ↔ ¬ ∅ ∈ (Pairs‘𝑉))
7	4, 5, 6	3imtr4i 292	. 2 ⊢ (∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} → ∅ ∉ (Pairs‘𝑉))
8	1, 7	ax-mp 5	1 ⊢ ∅ ∉ (Pairs‘𝑉)

Syntax hints:  ¬ wn 3   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ∉ wnel 3037  ∅c0 4287  {cpr 4584  ‘cfv 6500  Pairscspr 47837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-spr 47838

This theorem is referenced by: (None)