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Mirrors > Home > MPE Home > Th. List > elpqn | Structured version Visualization version GIF version |
Description: Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elpqn | ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq 9936 | . . 3 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} | |
2 | ssrab2 3836 | . . 3 ⊢ {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} ⊆ (N × N) | |
3 | 1, 2 | eqsstri 3784 | . 2 ⊢ Q ⊆ (N × N) |
4 | 3 | sseli 3748 | 1 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2145 ∀wral 3061 {crab 3065 class class class wbr 4786 × cxp 5247 ‘cfv 6031 2nd c2nd 7314 Ncnpi 9868 <N clti 9871 ~Q ceq 9875 Qcnq 9876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rab 3070 df-in 3730 df-ss 3737 df-nq 9936 |
This theorem is referenced by: nqereu 9953 nqerid 9957 enqeq 9958 addpqnq 9962 mulpqnq 9965 ordpinq 9967 addclnq 9969 mulclnq 9971 addnqf 9972 mulnqf 9973 adderpq 9980 mulerpq 9981 addassnq 9982 mulassnq 9983 distrnq 9985 mulidnq 9987 recmulnq 9988 ltsonq 9993 lterpq 9994 ltanq 9995 ltmnq 9996 ltexnq 9999 archnq 10004 wuncn 10193 |
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