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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 9689 | . . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))) | |
2 | simpl2 1058 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴) | |
3 | 1, 2 | sylbi 206 | . 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴) |
4 | 0pss 3965 | . 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) | |
5 | 3, 4 | sylib 207 | 1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∀wal 1473 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊊ wpss 3541 ∅c0 3874 class class class wbr 4583 Qcnq 9553 <Q cltq 9559 Pcnp 9560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-np 9682 |
This theorem is referenced by: 0npr 9693 npomex 9697 genpn0 9704 prlem934 9734 ltaddpr 9735 prlem936 9748 reclem2pr 9749 suplem1pr 9753 |
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