Theorem n0lpligALT 30508

Description: Alternate version of n0lplig 30507 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 30505. (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 2734	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 30503	. . 3 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅))
3		noel 4288	. . . . . . 7 ⊢ ¬ 𝑎 ∈ ∅
4	3	pm2.21i 119	. . . . . 6 ⊢ (𝑎 ∈ ∅ → ∅ ∉ 𝐺)
5	4	3ad2ant2 1134	. . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
6	5	a1i 11	. . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺))
7	6	rexlimivv 3176	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
8	2, 7	syl 17	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺)
9		simpr 484	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺)
10	8, 9	pm2.61danel 3048	1 ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113   ≠ wne 2930   ∉ wnel 3034  ∃wrex 3058  ∅c0 4283  ∪ cuni 4861  Pligcplig 30498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-v 3440  df-dif 3902  df-ss 3916  df-nul 4284  df-uni 4862  df-plig 30499

This theorem is referenced by: (None)