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Mirrors > Home > MPE Home > Th. List > nsnlpligALT | Structured version Visualization version GIF version |
Description: Alternate version of nsnlplig 30265 using the predicate ∉ instead of ¬ ∈ and whose proof is shorter. (Contributed by AV, 5-Dec-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nsnlpligALT | ⊢ (𝐺 ∈ Plig → {𝐴} ∉ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺 | |
2 | 1 | l2p 30263 | . . 3 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) |
3 | elsni 4641 | . . . . . . . 8 ⊢ (𝑎 ∈ {𝐴} → 𝑎 = 𝐴) | |
4 | elsni 4641 | . . . . . . . 8 ⊢ (𝑏 ∈ {𝐴} → 𝑏 = 𝐴) | |
5 | eqtr3 2753 | . . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏) | |
6 | eqneqall 2946 | . . . . . . . . 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 595 | . . . . . . 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 1113 | . . . . 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 3194 | . . 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 3055 | 1 ⊢ (𝐺 ∈ Plig → {𝐴} ∉ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∉ wnel 3041 ∃wrex 3065 {csn 4624 ∪ cuni 4903 Pligcplig 30258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1087 df-tru 1537 df-ex 1775 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-v 3471 df-in 3951 df-ss 3961 df-sn 4625 df-uni 4904 df-plig 30259 |
This theorem is referenced by: (None) |
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