Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > genpss | Structured version Visualization version Unicode version |
Description: The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
genp.1 | |
genp.2 |
Ref | Expression |
---|---|
genpss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | genp.1 | . . . 4 | |
2 | genp.2 | . . . 4 | |
3 | 1, 2 | genpelv 9822 | . . 3 |
4 | elprnq 9813 | . . . . . . . 8 | |
5 | 4 | ex 450 | . . . . . . 7 |
6 | elprnq 9813 | . . . . . . . 8 | |
7 | 6 | ex 450 | . . . . . . 7 |
8 | 5, 7 | im2anan9 880 | . . . . . 6 |
9 | 2 | caovcl 6828 | . . . . . 6 |
10 | 8, 9 | syl6 35 | . . . . 5 |
11 | eleq1a 2696 | . . . . 5 | |
12 | 10, 11 | syl6 35 | . . . 4 |
13 | 12 | rexlimdvv 3037 | . . 3 |
14 | 3, 13 | sylbid 230 | . 2 |
15 | 14 | ssrdv 3609 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cab 2608 wrex 2913 wss 3574 (class class class)co 6650 cmpt2 6652 cnq 9674 cnp 9681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-ni 9694 df-nq 9734 df-np 9803 |
This theorem is referenced by: genpcl 9830 |
Copyright terms: Public domain | W3C validator |