Theorem nsnlpligALT 30881

Description: Alternate version of nsnlplig 30880 using the predicate ∉ instead of ¬ ∈ and whose proof is shorter. (Contributed by AV, 5-Dec-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nsnlpligALT	⊢ (𝐺 ∈ Plig → {𝐴} ∉ 𝐺)

Proof of Theorem nsnlpligALT

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 30878	. . 3 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3		elsni 4608	. . . . . . . 8 ⊢ (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4		elsni 4608	. . . . . . . 8 ⊢ (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5		eqtr3 2787	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏)
6		eqneqall 2971	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → {𝐴} ∉ 𝐺))
7	5, 6	syl 18	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → {𝐴} ∉ 𝐺))
8	3, 4, 7	syl2an 608	. . . . . . 7 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎 ≠ 𝑏 → {𝐴} ∉ 𝐺))
9	8	impcom 413	. . . . . 6 ⊢ ((𝑎 ≠ 𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → {𝐴} ∉ 𝐺)
10	9	3impb 1132	. . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺)
11	10	a1i 11	. . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺))
12	11	rexlimivv 3209	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺)
13	2, 12	syl 18	. 2 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → {𝐴} ∉ 𝐺)
14		simpr 490	. 2 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∉ 𝐺) → {𝐴} ∉ 𝐺)
15	13, 14	pm2.61danel 3080	1 ⊢ (𝐺 ∈ Plig → {𝐴} ∉ 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∉ wnel 3066  ∃wrex 3091  {csn 4591  ∪ cuni 4874  Pligcplig 30873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-v 3459  df-ss 3923  df-sn 4592  df-uni 4875  df-plig 30874

This theorem is used by: (None)