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Theorem n0lpligALT 28256
 Description: Alternate version of n0lplig 28255 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 28253. (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 2824 . . . 4 𝐺 = 𝐺
21l2p 28251 . . 3 ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅))
3 noel 4278 . . . . . . 7 ¬ 𝑎 ∈ ∅
43pm2.21i 119 . . . . . 6 (𝑎 ∈ ∅ → ∅ ∉ 𝐺)
543ad2ant2 1131 . . . . 5 ((𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
65a1i 11 . . . 4 ((𝑎 𝐺𝑏 𝐺) → ((𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺))
76rexlimivv 3284 . . 3 (∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
82, 7syl 17 . 2 ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺)
9 simpr 488 . 2 ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺)
108, 9pm2.61danel 3131 1 (𝐺 ∈ Plig → ∅ ∉ 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2115   ≠ wne 3013   ∉ wnel 3117  ∃wrex 3133  ∅c0 4274  ∪ cuni 4819  Pligcplig 28246 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-v 3481  df-dif 3921  df-in 3925  df-ss 3935  df-nul 4275  df-uni 4820  df-plig 28247 This theorem is referenced by: (None)
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