Theorem n0lpligALT 31086

Description: Alternate version of n0lplig 31085 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 31083. (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 2760	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 31081	. . 3 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅))
3		noel 4283	. . . . . . 7 ⊢ ¬ 𝑎 ∈ ∅
4	3	pm2.21i 120	. . . . . 6 ⊢ (𝑎 ∈ ∅ → ∅ ∉ 𝐺)
5	4	3ad2ant2 1152	. . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
6	5	a1i 11	. . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺))
7	6	rexlimivv 3204	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
8	2, 7	syl 18	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺)
9		simpr 490	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺)
10	8, 9	pm2.61danel 3075	1 ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144   ≠ wne 2955   ∉ wnel 3061  ∃wrex 3086  ∅c0 4278  ∪ cuni 4866  Pligcplig 31076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-v 3452  df-ss 3896  df-dif 4047  df-nul 4279  df-uni 4867  df-plig 31077

This theorem is used by: (None)