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Mirrors > Home > MPE Home > Th. List > tcsni | Structured version Visualization version Unicode version |
Description: The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.) |
Ref | Expression |
---|---|
tc2.1 |
Ref | Expression |
---|---|
tcsni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tc2.1 | . . . . . 6 | |
2 | 1 | snss 4316 | . . . . 5 |
3 | 2 | anbi1i 731 | . . . 4 |
4 | 3 | abbii 2739 | . . 3 |
5 | 4 | inteqi 4479 | . 2 |
6 | 1 | tc2 8618 | . 2 |
7 | snex 4908 | . . 3 | |
8 | tcvalg 8614 | . . 3 | |
9 | 7, 8 | ax-mp 5 | . 2 |
10 | 5, 6, 9 | 3eqtr4ri 2655 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wcel 1990 cab 2608 cvv 3200 cun 3572 wss 3574 csn 4177 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: (None) |
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