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Theorem spr0el 42057
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.
StepHypRef Expression
1 spr0nelg 42051 . 2 ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
2 sprssspr 42056 . . . . 5 (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
32sseli 3632 . . . 4 (∅ ∈ (Pairs‘𝑉) → ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
43con3i 150 . . 3 (¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} → ¬ ∅ ∈ (Pairs‘𝑉))
5 df-nel 2927 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
6 df-nel 2927 . . 3 (∅ ∉ (Pairs‘𝑉) ↔ ¬ ∅ ∈ (Pairs‘𝑉))
74, 5, 63imtr4i 281 . 2 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} → ∅ ∉ (Pairs‘𝑉))
81, 7ax-mp 5 1 ∅ ∉ (Pairs‘𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wnel 2926  c0 3948  {cpr 4212  cfv 5926  Pairscspr 42052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-spr 42053
This theorem is referenced by: (None)
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