Theorem n0lpligALT 28263

Description: Alternate version of n0lplig 28262 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 28260. (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 2823	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 28258	. . 3 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅))
3		noel 4298	. . . . . . 7 ⊢ ¬ 𝑎 ∈ ∅
4	3	pm2.21i 119	. . . . . 6 ⊢ (𝑎 ∈ ∅ → ∅ ∉ 𝐺)
5	4	3ad2ant2 1130	. . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
6	5	a1i 11	. . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺))
7	6	rexlimivv 3294	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)
8	2, 7	syl 17	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺)
9		simpr 487	. 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺)
10	8, 9	pm2.61danel 3139	1 ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114   ≠ wne 3018   ∉ wnel 3125  ∃wrex 3141  ∅c0 4293  ∪ cuni 4840  Pligcplig 28253

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-nul 4294  df-uni 4841  df-plig 28254

This theorem is referenced by: (None)