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Mirrors > Home > MPE Home > Th. List > nqex | Structured version Visualization version GIF version |
Description: The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nqex | ⊢ Q ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq 9613 | . 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} | |
2 | niex 9582 | . . 3 ⊢ N ∈ V | |
3 | 2, 2 | xpex 6860 | . 2 ⊢ (N × N) ∈ V |
4 | 1, 3 | rabex2 4742 | 1 ⊢ Q ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 class class class wbr 4583 × cxp 5036 ‘cfv 5804 2nd c2nd 7058 Ncnpi 9545 <N clti 9548 ~Q ceq 9552 Qcnq 9553 |
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 |
This theorem is referenced by: npex 9687 elnp 9688 genpv 9700 genpdm 9703 |
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