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| Mirrors > Home > NFE Home > Th. List > enpw1pw | Unicode version | ||
| Description: Unit power class and power class commute within equivalence. Theorem XI.1.35 of [Rosser] p. 368. (Contributed by SF, 26-Feb-2015.) |
| Ref | Expression |
|---|---|
| enpw1pw.1 |
    |
| Ref | Expression |
|---|---|
| enpw1pw |
 1  1 |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pw1fnf1o 5855 |
. . . . 5
Pw1Fn  1c 1c | |
| 2 | f1of1 5286 |
. . . . 5
 Pw1Fn 1c1c  Pw1Fn 1c 1c | |
| 3 | 1, 2 | ax-mp 8 |
. . . 4
Pw1Fn 1c1c |
| 4 | pw1ss1c 4158 |
. . . 4
1  1c | |
| 5 | f1ores 5300 |
. . . 4
Pw1Fn 1c1c
![](wedge.gif "/\"){kind=small} 1 1c Pw1Fn  1 1 Pw1Fn  1 | |
| 6 | 3, 4, 5 | mp2an 653 |
. . 3
Pw1Fn 1 1 Pw1Fn 1 |
| 7 | df-ima 4727 |
. . . . 5
Pw1Fn 1  
  1 Pw1Fn  | |
| 8 | vex 2862 |
. . . . . . . . 9
| |
| 9 | 8 | elpw 3728 |
. . . . . . . 8
1  1 |
| 10 | 8 | sspw1 4335 |
. . . . . . . 8
1
1 |
| 11 | df-rex 2620 |
. . . . . . . . 9
1
1 | |
| 12 | df-pw 3724 |
. . . . . . . . . . . 12
| |
| 13 | 12 | abeq2i 2460 |
. . . . . . . . . . 11
|
| 14 | 13 | anbi1i 676 |
. . . . . . . . . 10
1 1 |
| 15 | 14 | exbii 1582 |
. . . . . . . . 9
1
1 |
| 16 | 11, 15 | bitr2i 241 |
. . . . . . . 8
1 1 |
| 17 | 9, 10, 16 | 3bitri 262 |
. . . . . . 7
1 1 |
| 18 | df-rex 2620 |
. . . . . . . 8
1 Pw1Fn 1
Pw1Fn | |
| 19 | elpw1 4144 |
. . . . . . . . . . 11
1 | |
| 20 | 19 | anbi1i 676 |
. . . . . . . . . 10
1
Pw1Fn
Pw1Fn |
| 21 | r19.41v 2764 |
. . . . . . . . . 10
Pw1Fn
Pw1Fn | |
| 22 | 20, 21 | bitr4i 243 |
. . . . . . . . 9
1
Pw1Fn
Pw1Fn |
| 23 | 22 | exbii 1582 |
. . . . . . . 8
1 Pw1Fn
Pw1Fn |
| 24 | rexcom4 2878 |
. . . . . . . . 9
Pw1Fn
Pw1Fn | |
| 25 | snex 4111 |
. . . . . . . . . . . 12
| |
| 26 | breq1 4642 |
. . . . . . . . . . . 12
Pw1Fn Pw1Fn | |
| 27 | 25, 26 | ceqsexv 2894 |
. . . . . . . . . . 11
Pw1Fn Pw1Fn |
| 28 | vex 2862 |
. . . . . . . . . . . 12
| |
| 29 | 28 | brpw1fn 5854 |
. . . . . . . . . . 11
Pw1Fn 1 |
| 30 | 27, 29 | bitri 240 |
. . . . . . . . . 10
Pw1Fn 1 |
| 31 | 30 | rexbii 2639 |
. . . . . . . . 9
Pw1Fn
1 |
| 32 | 24, 31 | bitr3i 242 |
. . . . . . . 8
Pw1Fn 1 |
| 33 | 18, 23, 32 | 3bitri 262 |
. . . . . . 7
1 Pw1Fn
1 |
| 34 | 17, 33 | bitr4i 243 |
. . . . . 6
1 1 Pw1Fn |
| 35 | 34 | abbi2i 2464 |
. . . . 5
1
1 Pw1Fn |
| 36 | 7, 35 | eqtr4i 2376 |
. . . 4
Pw1Fn 1 1 |
| 37 | f1oeq3 5283 |
. . . 4
Pw1Fn 1
1 Pw1Fn 1 1 Pw1Fn 1 Pw1Fn 1 1 1 | |
| 38 | 36, 37 | ax-mp 8 |
. . 3
Pw1Fn 1 1 Pw1Fn 1 Pw1Fn 1 1 1 |
| 39 | 6, 38 | mpbi 199 |
. 2
Pw1Fn 1 1 1 |
| 40 | pw1fnex 5852 |
. . . 4
Pw1Fn | |
| 41 | enpw1pw.1 |
. . . . . 6
| |
| 42 | 41 | pwex 4329 |
. . . . 5
|
| 43 | 42 | pw1ex 4303 |
. . . 4
1 |
| 44 | 40, 43 | resex 5117 |
. . 3
Pw1Fn 1 |
| 45 | 44 | f1oen 6033 |
. 2
Pw1Fn 1 1 1 1 1 |
| 46 | 39, 45 | ax-mp 8 |
1
1 1 |
| Colors of variables: wff setvar class |
| Syntax hints: wb 176
wa 358
wex 1541
wceq 1642
wcel 1710
cab 2339
wrex 2615
cvv 2859
wss 3257
cpw 3722
csn 3737
1cc1c 4134 1 cpw1 4135 class class class wbr 4639
cima 4722
cres 4774
wf1 4778
wf1o 4780 Pw1Fn cpw1fn 5765 cen 6028 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
| This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-mpt 5652 df-txp 5736 df-ins2 5750 df-ins3 5752 df-pw1fn 5766 df-en 6029 |
| This theorem is referenced by: ncpwpw1 6153 |
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