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 38640
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 7607 . 2 3𝑜 = suc 2𝑜
2 df2o2 7619 . . . 4 2𝑜 = {∅, {∅}}
32sneqi 4221 . . . 4 {2𝑜} = {{∅, {∅}}}
42, 3uneq12i 3798 . . 3 (2𝑜 ∪ {2𝑜}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5 df-suc 5767 . . 3 suc 2𝑜 = (2𝑜 ∪ {2𝑜})
6 df-tp 4215 . . 3 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
74, 5, 63eqtr4i 2683 . 2 suc 2𝑜 = {∅, {∅}, {∅, {∅}}}
81, 7eqtri 2673 1 3𝑜 = {∅, {∅}, {∅, {∅}}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cun 3605  c0 3948  {csn 4210  {cpr 4212  {ctp 4214  suc csuc 5763  2𝑜c2o 7599  3𝑜c3o 7600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215  df-suc 5767  df-1o 7605  df-2o 7606  df-3o 7607
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator