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Mirrors > Home > NFE Home > Th. List > nenpw1pwlem1 | Unicode version |
Description: Lemma for nenpw1pw 6086. Set up stratification. (Contributed by SF, 10-Mar-2015.) |
Ref | Expression |
---|---|
nenpw1pwlem1.1 |
Ref | Expression |
---|---|
nenpw1pwlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nenpw1pwlem1.1 | . . 3 | |
2 | dfrab2 3530 | . . 3 | |
3 | 1, 2 | eqtri 2373 | . 2 |
4 | vex 2862 | . . . . . . 7 | |
5 | 4 | elcompl 3225 | . . . . . 6 ∼ ⋃1 FullFun S ⋃1 FullFun S |
6 | eldm2 4899 | . . . . . . . 8 FullFun S FullFun S | |
7 | elin 3219 | . . . . . . . . . 10 FullFun S FullFun S | |
8 | snex 4111 | . . . . . . . . . . . . 13 | |
9 | 8 | brfullfun 5866 | . . . . . . . . . . . 12 FullFun |
10 | df-br 4640 | . . . . . . . . . . . 12 FullFun FullFun | |
11 | eqcom 2355 | . . . . . . . . . . . 12 | |
12 | 9, 10, 11 | 3bitr3i 266 | . . . . . . . . . . 11 FullFun |
13 | vex 2862 | . . . . . . . . . . . 12 | |
14 | 4, 13 | opelssetsn 4760 | . . . . . . . . . . 11 S |
15 | 12, 14 | anbi12i 678 | . . . . . . . . . 10 FullFun S |
16 | 7, 15 | bitri 240 | . . . . . . . . 9 FullFun S |
17 | 16 | exbii 1582 | . . . . . . . 8 FullFun S |
18 | 6, 17 | bitri 240 | . . . . . . 7 FullFun S |
19 | 4 | eluni1 4173 | . . . . . . 7 ⋃1 FullFun S FullFun S |
20 | fvex 5339 | . . . . . . . 8 | |
21 | 20 | clel3 2977 | . . . . . . 7 |
22 | 18, 19, 21 | 3bitr4i 268 | . . . . . 6 ⋃1 FullFun S |
23 | 5, 22 | xchbinx 301 | . . . . 5 ∼ ⋃1 FullFun S |
24 | 23 | abbi2i 2464 | . . . 4 ∼ ⋃1 FullFun S |
25 | vex 2862 | . . . . . . . . 9 | |
26 | 25 | fullfunex 5860 | . . . . . . . 8 FullFun |
27 | ssetex 4744 | . . . . . . . 8 S | |
28 | 26, 27 | inex 4105 | . . . . . . 7 FullFun S |
29 | 28 | dmex 5106 | . . . . . 6 FullFun S |
30 | 29 | uni1ex 4293 | . . . . 5 ⋃1 FullFun S |
31 | 30 | complex 4104 | . . . 4 ∼ ⋃1 FullFun S |
32 | 24, 31 | eqeltrri 2424 | . . 3 |
33 | inexg 4100 | . . 3 | |
34 | 32, 33 | mpan 651 | . 2 |
35 | 3, 34 | syl5eqel 2437 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 358 wex 1541 wceq 1642 wcel 1710 cab 2339 crab 2618 cvv 2859 ∼ ccompl 3205 cin 3208 csn 3737 ⋃1cuni1 4133 cop 4561 class class class wbr 4639 S csset 4719 cdm 4772 cfv 4781 FullFun cfullfun 5767 |
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-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-fv 4795 df-2nd 4797 df-fullfun 5768 |
This theorem is referenced by: nenpw1pwlem2 6085 |
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