| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11028 | . . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))) | |
| 2 | simpl2 1193 | . . 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 4447 | . 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 1087 ∀wal 1538 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊊ wpss 3952 ∅c0 4333 class class class wbr 5143 Qcnq 10892 <Q cltq 10898 Pcnp 10899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-pss 3971 df-nul 4334 df-np 11021 |
| This theorem is referenced by: 0npr 11032 npomex 11036 genpn0 11043 prlem934 11073 ltaddpr 11074 prlem936 11087 reclem2pr 11088 suplem1pr 11092 |
| Copyright terms: Public domain | W3C validator |