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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 10537 | . . 3 ⊢ Q ∈ V | |
2 | 1 | pwex 5273 | . 2 ⊢ 𝒫 Q ∈ V |
3 | pssss 4010 | . . . . 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 3980 | . . 3 ⊢ {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥 ∣ 𝑥 ⊆ Q} |
6 | df-np 10595 | . . 3 ⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} | |
7 | df-pw 4515 | . . 3 ⊢ 𝒫 Q = {𝑥 ∣ 𝑥 ⊆ Q} | |
8 | 5, 6, 7 | 3sstr4i 3944 | . 2 ⊢ P ⊆ 𝒫 Q |
9 | 2, 8 | ssexi 5215 | 1 ⊢ P ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 ∈ wcel 2110 {cab 2714 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ⊆ wss 3866 ⊊ wpss 3867 ∅c0 4237 𝒫 cpw 4513 class class class wbr 5053 Qcnq 10466 <Q cltq 10472 Pcnp 10473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-om 7645 df-ni 10486 df-nq 10526 df-np 10595 |
This theorem is referenced by: nrex1 10678 enrex 10681 |
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