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Mirrors > Home > MPE Home > Th. List > n0lpligALT | Structured version Visualization version GIF version |
Description: Alternate version of n0lplig 27335 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 27333. (Contributed by AV, 28-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
n0lpligALT | ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺 | |
2 | 1 | l2p 27331 | . . 3 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅)) |
3 | noel 3919 | . . . . . . 7 ⊢ ¬ 𝑎 ∈ ∅ | |
4 | 3 | pm2.21i 116 | . . . . . 6 ⊢ (𝑎 ∈ ∅ → ∅ ∉ 𝐺) |
5 | 4 | 3ad2ant2 1083 | . . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺) |
6 | 5 | a1i 11 | . . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)) |
7 | 6 | rexlimivv 3036 | . . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺) |
8 | 2, 7 | syl 17 | . 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺) |
9 | simpr 477 | . 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺) | |
10 | 8, 9 | pm2.61danel 2911 | 1 ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∃wrex 2913 ∅c0 3915 ∪ cuni 4436 Pligcplig 27326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-v 3202 df-dif 3577 df-nul 3916 df-uni 4437 df-plig 27327 |
This theorem is referenced by: (None) |
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