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Mirrors > Home > MPE Home > Th. List > npex | Structured version Visualization version GIF version |
Description: The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
npex | ⊢ P ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqex 10917 | . . 3 ⊢ Q ∈ V | |
2 | 1 | pwex 5378 | . 2 ⊢ 𝒫 Q ∈ V |
3 | pssss 4095 | . . . . 5 ⊢ (𝑥 ⊊ Q → 𝑥 ⊆ Q) | |
4 | 3 | ad2antlr 725 | . . . 4 ⊢ (((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧)) → 𝑥 ⊆ Q) |
5 | 4 | ss2abi 4063 | . . 3 ⊢ {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥 ∣ 𝑥 ⊆ Q} |
6 | df-np 10975 | . . 3 ⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} | |
7 | df-pw 4604 | . . 3 ⊢ 𝒫 Q = {𝑥 ∣ 𝑥 ⊆ Q} | |
8 | 5, 6, 7 | 3sstr4i 4025 | . 2 ⊢ P ⊆ 𝒫 Q |
9 | 2, 8 | ssexi 5322 | 1 ⊢ P ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 ∈ wcel 2106 {cab 2709 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⊆ wss 3948 ⊊ wpss 3949 ∅c0 4322 𝒫 cpw 4602 class class class wbr 5148 Qcnq 10846 <Q cltq 10852 Pcnp 10853 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-om 7855 df-ni 10866 df-nq 10906 df-np 10975 |
This theorem is referenced by: nrex1 11058 enrex 11061 |
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