Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tc2 | Structured version Visualization version Unicode version |
Description: A variant of the definition of the transitive closure function, using instead the smallest transitive set containing as a member, gives almost the same set, except that itself must be added because it is not usually a member of (and it is never a member if is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.) |
Ref | Expression |
---|---|
tc2.1 |
Ref | Expression |
---|---|
tc2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tc2.1 | . . . . 5 | |
2 | tcvalg 8614 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | trss 4761 | . . . . . . 7 | |
5 | 4 | imdistanri 727 | . . . . . 6 |
6 | 5 | ss2abi 3674 | . . . . 5 |
7 | intss 4498 | . . . . 5 | |
8 | 6, 7 | ax-mp 5 | . . . 4 |
9 | 3, 8 | eqsstri 3635 | . . 3 |
10 | 1 | elintab 4487 | . . . . 5 |
11 | simpl 473 | . . . . 5 | |
12 | 10, 11 | mpgbir 1726 | . . . 4 |
13 | 1 | snss 4316 | . . . 4 |
14 | 12, 13 | mpbi 220 | . . 3 |
15 | 9, 14 | unssi 3788 | . 2 |
16 | 1 | snid 4208 | . . . . 5 |
17 | elun2 3781 | . . . . 5 | |
18 | 16, 17 | ax-mp 5 | . . . 4 |
19 | uniun 4456 | . . . . . . 7 | |
20 | tctr 8616 | . . . . . . . . 9 | |
21 | df-tr 4753 | . . . . . . . . 9 | |
22 | 20, 21 | mpbi 220 | . . . . . . . 8 |
23 | 1 | unisn 4451 | . . . . . . . . 9 |
24 | tcid 8615 | . . . . . . . . . 10 | |
25 | 1, 24 | ax-mp 5 | . . . . . . . . 9 |
26 | 23, 25 | eqsstri 3635 | . . . . . . . 8 |
27 | 22, 26 | unssi 3788 | . . . . . . 7 |
28 | 19, 27 | eqsstri 3635 | . . . . . 6 |
29 | ssun1 3776 | . . . . . 6 | |
30 | 28, 29 | sstri 3612 | . . . . 5 |
31 | df-tr 4753 | . . . . 5 | |
32 | 30, 31 | mpbir 221 | . . . 4 |
33 | fvex 6201 | . . . . . 6 | |
34 | snex 4908 | . . . . . 6 | |
35 | 33, 34 | unex 6956 | . . . . 5 |
36 | eleq2 2690 | . . . . . 6 | |
37 | treq 4758 | . . . . . 6 | |
38 | 36, 37 | anbi12d 747 | . . . . 5 |
39 | 35, 38 | elab 3350 | . . . 4 |
40 | 18, 32, 39 | mpbir2an 955 | . . 3 |
41 | intss1 4492 | . . 3 | |
42 | 40, 41 | ax-mp 5 | . 2 |
43 | 15, 42 | eqssi 3619 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cab 2608 cvv 3200 cun 3572 wss 3574 csn 4177 cuni 4436 cint 4475 wtr 4752 cfv 5888 ctc 8612 |
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-rep 4771 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-reu 2919 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-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-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 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-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-tc 8613 |
This theorem is referenced by: tcsni 8619 |
Copyright terms: Public domain | W3C validator |