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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 10961 | . . 3 ⊢ Q ∈ V | |
2 | 1 | pwex 5386 | . 2 ⊢ 𝒫 Q ∈ V |
3 | pssss 4108 | . . . . 5 ⊢ (𝑥 ⊊ Q → 𝑥 ⊆ Q) | |
4 | 3 | ad2antlr 727 | . . . 4 ⊢ (((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧)) → 𝑥 ⊆ Q) |
5 | 4 | ss2abi 4077 | . . 3 ⊢ {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥 ∣ 𝑥 ⊆ Q} |
6 | df-np 11019 | . . 3 ⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} | |
7 | df-pw 4607 | . . 3 ⊢ 𝒫 Q = {𝑥 ∣ 𝑥 ⊆ Q} | |
8 | 5, 6, 7 | 3sstr4i 4039 | . 2 ⊢ P ⊆ 𝒫 Q |
9 | 2, 8 | ssexi 5328 | 1 ⊢ P ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 ⊊ wpss 3964 ∅c0 4339 𝒫 cpw 4605 class class class wbr 5148 Qcnq 10890 <Q cltq 10896 Pcnp 10897 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-om 7888 df-ni 10910 df-nq 10950 df-np 11019 |
This theorem is referenced by: nrex1 11102 enrex 11105 |
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