Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df3o3 Structured version   Visualization version   GIF version

Theorem df3o3 40242
Description: Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.)
Assertion
Ref Expression
df3o3 3o = {∅, {∅}, {∅, {∅}}}

Proof of Theorem df3o3
StepHypRef Expression
1 df-3o 8098 . 2 3o = suc 2o
2 df2o2 8112 . . . 4 2o = {∅, {∅}}
32sneqi 4574 . . . 4 {2o} = {{∅, {∅}}}
42, 3uneq12i 4140 . . 3 (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5 df-suc 6194 . . 3 suc 2o = (2o ∪ {2o})
6 df-tp 4568 . . 3 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
74, 5, 63eqtr4i 2858 . 2 suc 2o = {∅, {∅}, {∅, {∅}}}
81, 7eqtri 2848 1 3o = {∅, {∅}, {∅, {∅}}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  cun 3937  c0 4294  {csn 4563  {cpr 4565  {ctp 4567  suc csuc 6190  2oc2o 8090  3oc3o 8091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4564  df-pr 4566  df-tp 4568  df-suc 6194  df-1o 8096  df-2o 8097  df-3o 8098
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator