Theorem n0lpligALT 30847

Description: Alternate version of n0lplig 30846 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 30844. (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.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 30842	. . 3 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅))
3		noel 4290	. . . . . . 7 ⊢ ¬ 𝑎 ∈ ∅
4	3	pm2.21i 120	. . . . . 6 ⊢ (𝑎 ∈ ∅ → ∅ ∉ 𝐺)
5	4	3ad2ant2 1151	. . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
6	5	a1i 11	. . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺))
7	6	rexlimivv 3206	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
8	2, 7	syl 18	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺)
9		simpr 489	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺)
10	8, 9	pm2.61danel 3077	1 ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   ∈ wcel 2142   ≠ wne 2957   ∉ wnel 3063  ∃wrex 3088  ∅c0 4285  ∪ cuni 4871  Pligcplig 30837

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286  df-uni 4872  df-plig 30838

This theorem is used by: (None)