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Mirrors > Home > MPE Home > Th. List > rexanuz | Structured version Visualization version Unicode version |
Description: Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.) |
Ref | Expression |
---|---|
rexanuz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3064 | . . . 4 | |
2 | 1 | rexbii 3041 | . . 3 |
3 | r19.40 3088 | . . 3 | |
4 | 2, 3 | sylbi 207 | . 2 |
5 | uzf 11690 | . . . 4 | |
6 | ffn 6045 | . . . 4 | |
7 | raleq 3138 | . . . . 5 | |
8 | 7 | rexrn 6361 | . . . 4 |
9 | 5, 6, 8 | mp2b 10 | . . 3 |
10 | raleq 3138 | . . . . 5 | |
11 | 10 | rexrn 6361 | . . . 4 |
12 | 5, 6, 11 | mp2b 10 | . . 3 |
13 | uzin2 14084 | . . . . . . . . 9 | |
14 | inss1 3833 | . . . . . . . . . . . 12 | |
15 | ssralv 3666 | . . . . . . . . . . . 12 | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . . . 11 |
17 | inss2 3834 | . . . . . . . . . . . 12 | |
18 | ssralv 3666 | . . . . . . . . . . . 12 | |
19 | 17, 18 | ax-mp 5 | . . . . . . . . . . 11 |
20 | 16, 19 | anim12i 590 | . . . . . . . . . 10 |
21 | r19.26 3064 | . . . . . . . . . 10 | |
22 | 20, 21 | sylibr 224 | . . . . . . . . 9 |
23 | raleq 3138 | . . . . . . . . . 10 | |
24 | 23 | rspcev 3309 | . . . . . . . . 9 |
25 | 13, 22, 24 | syl2an 494 | . . . . . . . 8 |
26 | 25 | an4s 869 | . . . . . . 7 |
27 | 26 | rexlimdvaa 3032 | . . . . . 6 |
28 | 27 | rexlimiva 3028 | . . . . 5 |
29 | 28 | imp 445 | . . . 4 |
30 | raleq 3138 | . . . . . 6 | |
31 | 30 | rexrn 6361 | . . . . 5 |
32 | 5, 6, 31 | mp2b 10 | . . . 4 |
33 | 29, 32 | sylib 208 | . . 3 |
34 | 9, 12, 33 | syl2anbr 497 | . 2 |
35 | 4, 34 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wcel 1990 wral 2912 wrex 2913 cin 3573 wss 3574 cpw 4158 crn 5115 wfn 5883 wf 5884 cfv 5888 cz 11377 cuz 11687 |
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-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
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-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-neg 10269 df-z 11378 df-uz 11688 |
This theorem is referenced by: rexfiuz 14087 rexuz3 14088 rexanuz2 14089 |
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