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Theorem n0lpligALT 29468
Description: Alternate version of n0lplig 29467 using the predicate instead of ¬ ∈ and whose proof bypasses nsnlplig 29465. (Contributed by AV, 28-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
n0lpligALT (𝐺 ∈ Plig → ∅ ∉ 𝐺)

Proof of Theorem n0lpligALT
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 𝐺 = 𝐺
21l2p 29463 . . 3 ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅))
3 noel 4291 . . . . . . 7 ¬ 𝑎 ∈ ∅
43pm2.21i 119 . . . . . 6 (𝑎 ∈ ∅ → ∅ ∉ 𝐺)
543ad2ant2 1135 . . . . 5 ((𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
65a1i 11 . . . 4 ((𝑎 𝐺𝑏 𝐺) → ((𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺))
76rexlimivv 3193 . . 3 (∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
82, 7syl 17 . 2 ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺)
9 simpr 486 . 2 ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺)
108, 9pm2.61danel 3060 1 (𝐺 ∈ Plig → ∅ ∉ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wne 2940  wnel 3046  wrex 3070  c0 4283   cuni 4866  Pligcplig 29458
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284  df-uni 4867  df-plig 29459
This theorem is referenced by: (None)
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