Theorem nsnlpligALT 30448

Description: Alternate version of nsnlplig 30447 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 2734	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 30445	. . 3 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3		elsni 4625	. . . . . . . 8 ⊢ (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4		elsni 4625	. . . . . . . 8 ⊢ (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5		eqtr3 2756	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏)
6		eqneqall 2942	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → {𝐴} ∉ 𝐺))
7	5, 6	syl 17	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → {𝐴} ∉ 𝐺))
8	3, 4, 7	syl2an 596	. . . . . . 7 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎 ≠ 𝑏 → {𝐴} ∉ 𝐺))
9	8	impcom 407	. . . . . 6 ⊢ ((𝑎 ≠ 𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → {𝐴} ∉ 𝐺)
10	9	3impb 1114	. . . . 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 3188	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺)
13	2, 12	syl 17	. 2 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → {𝐴} ∉ 𝐺)
14		simpr 484	. 2 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∉ 𝐺) → {𝐴} ∉ 𝐺)
15	13, 14	pm2.61danel 3049	1 ⊢ (𝐺 ∈ Plig → {𝐴} ∉ 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2931   ∉ wnel 3035  ∃wrex 3059  {csn 4608  ∪ cuni 4889  Pligcplig 30440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-v 3466  df-ss 3950  df-sn 4609  df-uni 4890  df-plig 30441

This theorem is referenced by: (None)