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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 10409 | . . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))) | |
2 | simpl2 1188 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴) | |
3 | 1, 2 | sylbi 219 | . 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴) |
4 | 0pss 4395 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) | |
5 | 3, 4 | sylib 220 | 1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∀wal 1531 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊊ wpss 3936 ∅c0 4290 class class class wbr 5065 Qcnq 10273 <Q cltq 10279 Pcnp 10280 |
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-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-np 10402 |
This theorem is referenced by: 0npr 10413 npomex 10417 genpn0 10424 prlem934 10454 ltaddpr 10455 prlem936 10468 reclem2pr 10469 suplem1pr 10473 |
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