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| 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 10963 | . . 3 ⊢ Q ∈ V | |
| 2 | 1 | pwex 5380 | . 2 ⊢ 𝒫 Q ∈ V | 
| 3 | pssss 4098 | . . . . 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 4067 | . . 3 ⊢ {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥 ∣ 𝑥 ⊆ Q} | 
| 6 | df-np 11021 | . . 3 ⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} | |
| 7 | df-pw 4602 | . . 3 ⊢ 𝒫 Q = {𝑥 ∣ 𝑥 ⊆ Q} | |
| 8 | 5, 6, 7 | 3sstr4i 4035 | . 2 ⊢ P ⊆ 𝒫 Q | 
| 9 | 2, 8 | ssexi 5322 | 1 ⊢ P ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ⊊ wpss 3952 ∅c0 4333 𝒫 cpw 4600 class class class wbr 5143 Qcnq 10892 <Q cltq 10898 Pcnp 10899 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-om 7888 df-ni 10912 df-nq 10952 df-np 11021 | 
| This theorem is referenced by: nrex1 11104 enrex 11107 | 
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