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Mirrors > Home > MPE Home > Th. List > tcsni | Structured version Visualization version GIF version |
Description: The transitive closure of a singleton. Proof suggested by GΓ©rard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.) |
Ref | Expression |
---|---|
tc2.1 | β’ π΄ β V |
Ref | Expression |
---|---|
tcsni | β’ (TCβ{π΄}) = ((TCβπ΄) βͺ {π΄}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tc2.1 | . . . . . 6 β’ π΄ β V | |
2 | 1 | snss 4751 | . . . . 5 β’ (π΄ β π₯ β {π΄} β π₯) |
3 | 2 | anbi1i 625 | . . . 4 β’ ((π΄ β π₯ β§ Tr π₯) β ({π΄} β π₯ β§ Tr π₯)) |
4 | 3 | abbii 2807 | . . 3 β’ {π₯ β£ (π΄ β π₯ β§ Tr π₯)} = {π₯ β£ ({π΄} β π₯ β§ Tr π₯)} |
5 | 4 | inteqi 4916 | . 2 β’ β© {π₯ β£ (π΄ β π₯ β§ Tr π₯)} = β© {π₯ β£ ({π΄} β π₯ β§ Tr π₯)} |
6 | 1 | tc2 9685 | . 2 β’ ((TCβπ΄) βͺ {π΄}) = β© {π₯ β£ (π΄ β π₯ β§ Tr π₯)} |
7 | snex 5393 | . . 3 β’ {π΄} β V | |
8 | tcvalg 9681 | . . 3 β’ ({π΄} β V β (TCβ{π΄}) = β© {π₯ β£ ({π΄} β π₯ β§ Tr π₯)}) | |
9 | 7, 8 | ax-mp 5 | . 2 β’ (TCβ{π΄}) = β© {π₯ β£ ({π΄} β π₯ β§ Tr π₯)} |
10 | 5, 6, 9 | 3eqtr4ri 2776 | 1 β’ (TCβ{π΄}) = ((TCβπ΄) βͺ {π΄}) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 {cab 2714 Vcvv 3448 βͺ cun 3913 β wss 3915 {csn 4591 β© cint 4912 Tr wtr 5227 βcfv 6501 TCctc 9679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-tc 9680 |
This theorem is referenced by: (None) |
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