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Mirrors > Home > MPE Home > Th. List > prn0 | Structured version Visualization version GIF version |
Description: A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prn0 | ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnpi 11026 | . . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))) | |
2 | simpl2 1191 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴) | |
3 | 1, 2 | sylbi 217 | . 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴) |
4 | 0pss 4453 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) | |
5 | 3, 4 | sylib 218 | 1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wal 1535 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ⊊ wpss 3964 ∅c0 4339 class class class wbr 5148 Qcnq 10890 <Q cltq 10896 Pcnp 10897 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-ss 3980 df-pss 3983 df-nul 4340 df-np 11019 |
This theorem is referenced by: 0npr 11030 npomex 11034 genpn0 11041 prlem934 11071 ltaddpr 11072 prlem936 11085 reclem2pr 11086 suplem1pr 11090 |
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