Theorem n0lpligALT 30004

Description: Alternate version of n0lplig 30003 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 30001. (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 2730	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 29999	. . 3 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅))
3		noel 4329	. . . . . . 7 ⊢ ¬ 𝑎 ∈ ∅
4	3	pm2.21i 119	. . . . . 6 ⊢ (𝑎 ∈ ∅ → ∅ ∉ 𝐺)
5	4	3ad2ant2 1132	. . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
6	5	a1i 11	. . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺))
7	6	rexlimivv 3197	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
8	2, 7	syl 17	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺)
9		simpr 483	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺)
10	8, 9	pm2.61danel 3058	1 ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   ∈ wcel 2104   ≠ wne 2938   ∉ wnel 3044  ∃wrex 3068  ∅c0 4321  ∪ cuni 4907  Pligcplig 29994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322  df-uni 4908  df-plig 29995

This theorem is referenced by: (None)