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Mirrors > Home > MPE Home > Th. List > 0nnq | Structured version Visualization version GIF version |
Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0nnq | ⊢ ¬ ∅ ∈ Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 5067 | . 2 ⊢ ¬ ∅ ∈ (N × N) | |
2 | df-nq 9613 | . . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} | |
3 | ssrab2 3650 | . . . 4 ⊢ {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} ⊆ (N × N) | |
4 | 2, 3 | eqsstri 3598 | . . 3 ⊢ Q ⊆ (N × N) |
5 | 4 | sseli 3564 | . 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N)) |
6 | 1, 5 | mto 187 | 1 ⊢ ¬ ∅ ∈ Q |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∅c0 3874 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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-nq 9613 |
This theorem is referenced by: adderpq 9657 mulerpq 9658 addassnq 9659 mulassnq 9660 distrnq 9662 recmulnq 9665 recclnq 9667 ltanq 9672 ltmnq 9673 ltexnq 9676 nsmallnq 9678 ltbtwnnq 9679 ltrnq 9680 prlem934 9734 ltaddpr 9735 ltexprlem2 9738 ltexprlem3 9739 ltexprlem4 9740 ltexprlem6 9742 ltexprlem7 9743 prlem936 9748 reclem2pr 9749 |
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