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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 10950 | . 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} | |
2 | niex 10919 | . . 3 ⊢ N ∈ V | |
3 | 2, 2 | xpex 7772 | . 2 ⊢ (N × N) ∈ V |
4 | 1, 3 | rabex2 5347 | 1 ⊢ Q ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 class class class wbr 5148 × cxp 5687 ‘cfv 6563 2nd c2nd 8012 Ncnpi 10882 <N clti 10885 ~Q ceq 10889 Qcnq 10890 |
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 |
This theorem is referenced by: npex 11024 elnp 11025 genpv 11037 genpdm 11040 |
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