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Theorem nsnlpligALT 29466
Description: Alternate version of nsnlplig 29465 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 2733 . . . 4 𝐺 = 𝐺
21l2p 29463 . . 3 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3 elsni 4604 . . . . . . . 8 (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4 elsni 4604 . . . . . . . 8 (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5 eqtr3 2759 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐴) → 𝑎 = 𝑏)
6 eqneqall 2951 . . . . . . . . 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 409 . . . . . 6 ((𝑎𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → {𝐴} ∉ 𝐺)
1093impb 1116 . . . . 5 ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺)
1110a1i 11 . . . 4 ((𝑎 𝐺𝑏 𝐺) → ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺))
1211rexlimivv 3193 . . 3 (∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → {𝐴} ∉ 𝐺)
132, 12syl 17 . 2 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → {𝐴} ∉ 𝐺)
14 simpr 486 . 2 ((𝐺 ∈ Plig ∧ {𝐴} ∉ 𝐺) → {𝐴} ∉ 𝐺)
1513, 14pm2.61danel 3060 1 (𝐺 ∈ Plig → {𝐴} ∉ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wnel 3046  wrex 3070  {csn 4587   cuni 4866  Pligcplig 29458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-v 3446  df-in 3918  df-ss 3928  df-sn 4588  df-uni 4867  df-plig 29459
This theorem is referenced by: (None)
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