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Theorem spr0el 42242
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 42236 . 2 ∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
2 sprssspr 42241 . . . . 5 (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
32sseli 3740 . . . 4 (∅ ∈ (Pairs‘𝑉) → ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
43con3i 150 . . 3 (¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} → ¬ ∅ ∈ (Pairs‘𝑉))
5 df-nel 3036 . . 3 (∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ¬ ∅ ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
6 df-nel 3036 . . 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 1632  wex 1853  wcel 2139  {cab 2746  wnel 3035  c0 4058  {cpr 4323  cfv 6049  Pairscspr 42237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-spr 42238
This theorem is referenced by: (None)
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