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Theorem spr0el 45764
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 45758 . 2 ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
2 sprssspr 45763 . . . . 5 (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
32sseli 3944 . . . 4 (∅ ∈ (Pairs‘𝑉) → ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
43con3i 154 . . 3 (¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} → ¬ ∅ ∈ (Pairs‘𝑉))
5 df-nel 3047 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
6 df-nel 3047 . . 3 (∅ ∉ (Pairs‘𝑉) ↔ ¬ ∅ ∈ (Pairs‘𝑉))
74, 5, 63imtr4i 292 . 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 1542  wex 1782  wcel 2107  {cab 2710  wnel 3046  c0 4286  {cpr 4592  cfv 6500  Pairscspr 45759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-spr 45760
This theorem is referenced by: (None)
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