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Theorem nsnlpligALT 28258
 Description: Alternate version of nsnlplig 28257 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.
StepHypRef Expression
1 eqid 2821 . . . 4 𝐺 = 𝐺
21l2p 28255 . . 3 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3 elsni 4583 . . . . . . . 8 (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4 elsni 4583 . . . . . . . 8 (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5 eqtr3 2843 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐴) → 𝑎 = 𝑏)
6 eqneqall 3027 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎𝑏 → {𝐴} ∉ 𝐺))
75, 6syl 17 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐴) → (𝑎𝑏 → {𝐴} ∉ 𝐺))
83, 4, 7syl2an 597 . . . . . . 7 ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎𝑏 → {𝐴} ∉ 𝐺))
98impcom 410 . . . . . 6 ((𝑎𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → {𝐴} ∉ 𝐺)
1093impb 1111 . . . . 5 ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺)
1110a1i 11 . . . 4 ((𝑎 𝐺𝑏 𝐺) → ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺))
1211rexlimivv 3292 . . 3 (∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺)
132, 12syl 17 . 2 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → {𝐴} ∉ 𝐺)
14 simpr 487 . 2 ((𝐺 ∈ Plig ∧ {𝐴} ∉ 𝐺) → {𝐴} ∉ 𝐺)
1513, 14pm2.61danel 3137 1 (𝐺 ∈ Plig → {𝐴} ∉ 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   ∉ wnel 3123  ∃wrex 3139  {csn 4566  ∪ cuni 4837  Pligcplig 28250 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-v 3496  df-in 3942  df-ss 3951  df-sn 4567  df-uni 4838  df-plig 28251 This theorem is referenced by: (None)
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