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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 9624 | . . 3 ⊢ Q ∈ V | |
2 | 1 | pwex 4774 | . 2 ⊢ 𝒫 Q ∈ V |
3 | pssss 3664 | . . . . 5 ⊢ (𝑥 ⊊ Q → 𝑥 ⊆ Q) | |
4 | 3 | ad2antlr 759 | . . . 4 ⊢ (((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧)) → 𝑥 ⊆ Q) |
5 | 4 | ss2abi 3637 | . . 3 ⊢ {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥 ∣ 𝑥 ⊆ Q} |
6 | df-np 9682 | . . 3 ⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} | |
7 | df-pw 4110 | . . 3 ⊢ 𝒫 Q = {𝑥 ∣ 𝑥 ⊆ Q} | |
8 | 5, 6, 7 | 3sstr4i 3607 | . 2 ⊢ P ⊆ 𝒫 Q |
9 | 2, 8 | ssexi 4731 | 1 ⊢ P ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ⊊ wpss 3541 ∅c0 3874 𝒫 cpw 4108 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 df-ni 9573 df-nq 9613 df-np 9682 |
This theorem is referenced by: enrex 9767 axcnex 9847 |
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