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Mirrors > Home > NFE Home > Th. List > pw1fnex | Unicode version |
Description: The unit power class function is a set. (Contributed by SF, 25-Feb-2015.) |
Ref | Expression |
---|---|
pw1fnex | Pw1Fn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pw1fn 5766 | . . 3 Pw1Fn 1c 1 | |
2 | oteltxp 5782 | . . . . . . . 8 SI S S 1c SI S S 1c | |
3 | snex 4111 | . . . . . . . . . . 11 | |
4 | 3 | ideq 4870 | . . . . . . . . . 10 |
5 | df-br 4640 | . . . . . . . . . 10 | |
6 | eqcom 2355 | . . . . . . . . . . 11 | |
7 | vex 2862 | . . . . . . . . . . . 12 | |
8 | 7 | sneqb 3876 | . . . . . . . . . . 11 |
9 | 6, 8 | bitri 240 | . . . . . . . . . 10 |
10 | 4, 5, 9 | 3bitr3i 266 | . . . . . . . . 9 |
11 | oteltxp 5782 | . . . . . . . . . . . 12 SI S S SI S S | |
12 | snex 4111 | . . . . . . . . . . . . . . 15 | |
13 | 7, 12 | brsnsi 5773 | . . . . . . . . . . . . . 14 SI S S |
14 | df-br 4640 | . . . . . . . . . . . . . 14 SI S SI S | |
15 | brcnv 4892 | . . . . . . . . . . . . . . 15 S S | |
16 | vex 2862 | . . . . . . . . . . . . . . . 16 | |
17 | 16, 7 | brssetsn 4759 | . . . . . . . . . . . . . . 15 S |
18 | 15, 17 | bitri 240 | . . . . . . . . . . . . . 14 S |
19 | 13, 14, 18 | 3bitr3i 266 | . . . . . . . . . . . . 13 SI S |
20 | vex 2862 | . . . . . . . . . . . . . 14 | |
21 | 7, 20 | opelssetsn 4760 | . . . . . . . . . . . . 13 S |
22 | 19, 21 | anbi12i 678 | . . . . . . . . . . . 12 SI S S |
23 | 11, 22 | bitri 240 | . . . . . . . . . . 11 SI S S |
24 | 23 | exbii 1582 | . . . . . . . . . 10 SI S S |
25 | elima1c 4947 | . . . . . . . . . 10 SI S S 1c SI S S | |
26 | eluni 3894 | . . . . . . . . . 10 | |
27 | 24, 25, 26 | 3bitr4i 268 | . . . . . . . . 9 SI S S 1c |
28 | 10, 27 | anbi12i 678 | . . . . . . . 8 SI S S 1c |
29 | 2, 28 | bitri 240 | . . . . . . 7 SI S S 1c |
30 | ancom 437 | . . . . . . 7 | |
31 | 29, 30 | bitri 240 | . . . . . 6 SI S S 1c |
32 | 31 | exbii 1582 | . . . . 5 SI S S 1c |
33 | elimapw11c 4948 | . . . . 5 SI S S 1c1 1c SI S S 1c | |
34 | elpw1 4144 | . . . . . 6 1 | |
35 | df-rex 2620 | . . . . . 6 | |
36 | 34, 35 | bitri 240 | . . . . 5 1 |
37 | 32, 33, 36 | 3bitr4i 268 | . . . 4 SI S S 1c1 1c 1 |
38 | 37 | releqmpt 5808 | . . 3 1c ∼ Ins3 S Ins2 SI S S 1c1 1c1c 1c 1 |
39 | 1, 38 | eqtr4i 2376 | . 2 Pw1Fn 1c ∼ Ins3 S Ins2 SI S S 1c1 1c1c |
40 | 1cex 4142 | . . 3 1c | |
41 | idex 5504 | . . . . 5 | |
42 | ssetex 4744 | . . . . . . . . 9 S | |
43 | 42 | cnvex 5102 | . . . . . . . 8 S |
44 | 43 | siex 4753 | . . . . . . 7 SI S |
45 | 44, 42 | txpex 5785 | . . . . . 6 SI S S |
46 | 45, 40 | imaex 4747 | . . . . 5 SI S S 1c |
47 | 41, 46 | txpex 5785 | . . . 4 SI S S 1c |
48 | 40 | pw1ex 4303 | . . . 4 1 1c |
49 | 47, 48 | imaex 4747 | . . 3 SI S S 1c1 1c |
50 | 40, 49 | mptexlem 5810 | . 2 1c ∼ Ins3 S Ins2 SI S S 1c1 1c1c |
51 | 39, 50 | eqeltri 2423 | 1 Pw1Fn |
Colors of variables: wff setvar class |
Syntax hints: wa 358 wex 1541 wceq 1642 wcel 1710 wrex 2615 cvv 2859 ∼ ccompl 3205 cin 3208 csymdif 3209 csn 3737 cuni 3891 1cc1c 4134 1 cpw1 4135 cop 4561 class class class wbr 4639 S csset 4719 SI csi 4720 cima 4722 cid 4763 cxp 4770 ccnv 4771 cmpt 5651 ctxp 5735 Ins2 cins2 5749 Ins3 cins3 5751 Pw1Fn cpw1fn 5765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-2nd 4797 df-mpt 5652 df-txp 5736 df-ins2 5750 df-ins3 5752 df-pw1fn 5766 |
This theorem is referenced by: enpw1pw 6075 ovcelem1 6171 ceex 6174 tcfnex 6244 |
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