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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 10399 | . . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))) | |
2 | simpl2 1189 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴) | |
3 | 1, 2 | sylbi 220 | . 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴) |
4 | 0pss 4352 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) | |
5 | 3, 4 | sylib 221 | 1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∀wal 1536 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⊊ wpss 3882 ∅c0 4243 class class class wbr 5030 Qcnq 10263 <Q cltq 10269 Pcnp 10270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-np 10392 |
This theorem is referenced by: 0npr 10403 npomex 10407 genpn0 10414 prlem934 10444 ltaddpr 10445 prlem936 10458 reclem2pr 10459 suplem1pr 10463 |
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