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Mirrors > Home > MPE Home > Th. List > n0lpligALT | Structured version Visualization version GIF version |
Description: Alternate version of n0lplig 28259 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 28257. (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 2821 | . . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺 | |
2 | 1 | l2p 28255 | . . 3 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅)) |
3 | noel 4295 | . . . . . . 7 ⊢ ¬ 𝑎 ∈ ∅ | |
4 | 3 | pm2.21i 119 | . . . . . 6 ⊢ (𝑎 ∈ ∅ → ∅ ∉ 𝐺) |
5 | 4 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺) |
6 | 5 | a1i 11 | . . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺)) |
7 | 6 | rexlimivv 3292 | . . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ ∅ ∧ 𝑏 ∈ ∅) → ∅ ∉ 𝐺) |
8 | 2, 7 | syl 17 | . 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∈ 𝐺) → ∅ ∉ 𝐺) |
9 | simpr 487 | . 2 ⊢ ((𝐺 ∈ Plig ∧ ∅ ∉ 𝐺) → ∅ ∉ 𝐺) | |
10 | 8, 9 | pm2.61danel 3137 | 1 ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 ≠ wne 3016 ∉ wnel 3123 ∃wrex 3139 ∅c0 4290 ∪ 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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-uni 4838 df-plig 28251 |
This theorem is referenced by: (None) |
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