![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4790 | . . . . 5 β’ (π΄ β π₯ β {π΄} β π₯) |
3 | 2 | anbi1i 623 | . . . 4 β’ ((π΄ β π₯ β§ Tr π₯) β ({π΄} β π₯ β§ Tr π₯)) |
4 | 3 | abbii 2798 | . . 3 β’ {π₯ β£ (π΄ β π₯ β§ Tr π₯)} = {π₯ β£ ({π΄} β π₯ β§ Tr π₯)} |
5 | 4 | inteqi 4953 | . 2 β’ β© {π₯ β£ (π΄ β π₯ β§ Tr π₯)} = β© {π₯ β£ ({π΄} β π₯ β§ Tr π₯)} |
6 | 1 | tc2 9766 | . 2 β’ ((TCβπ΄) βͺ {π΄}) = β© {π₯ β£ (π΄ β π₯ β§ Tr π₯)} |
7 | snex 5433 | . . 3 β’ {π΄} β V | |
8 | tcvalg 9762 | . . 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 2767 | 1 β’ (TCβ{π΄}) = ((TCβπ΄) βͺ {π΄}) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1534 β wcel 2099 {cab 2705 Vcvv 3471 βͺ cun 3945 β wss 3947 {csn 4629 β© cint 4949 Tr wtr 5265 βcfv 6548 TCctc 9760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-inf2 9665 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-tc 9761 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |