Theorem df2o3 6483

Description: Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
df2o3	⊢ 2o = {∅, 1o}

Proof of Theorem df2o3

Step	Hyp	Ref	Expression
1		df-2o 6470	. 2 ⊢ 2o = suc 1o
2		df-suc 4402	. 2 ⊢ suc 1o = (1o ∪ {1o})
3		df1o2 6482	. . . 4 ⊢ 1o = {∅}
4	3	uneq1i 3309	. . 3 ⊢ (1o ∪ {1o}) = ({∅} ∪ {1o})
5		df-pr 3625	. . 3 ⊢ {∅, 1o} = ({∅} ∪ {1o})
6	4, 5	eqtr4i 2217	. 2 ⊢ (1o ∪ {1o}) = {∅, 1o}
7	1, 2, 6	3eqtri 2218	1 ⊢ 2o = {∅, 1o}

Syntax hints:   = wceq 1364   ∪ cun 3151  ∅c0 3446  {csn 3618  {cpr 3619  suc csuc 4396  1_oc1o 6462  2_oc2o 6463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-nul 3447  df-pr 3625  df-suc 4402  df-1o 6469  df-2o 6470

This theorem is referenced by:  df2o2  6484  2oconcl  6492  0lt2o  6494  1lt2o  6495  el2oss1o  6496  en2eqpr  6963  nninfisol  7192  finomni  7199  exmidomniim  7200  exmidomni  7201  ismkvnex  7214  nninfwlpoimlemginf  7235  exmidfodomrlemr  7262  exmidfodomrlemrALT  7263  xp2dju  7275  pw1nel3  7291  sucpw1nel3  7293  nninfctlemfo  12177  unct  12599  fnpr2o  12922  fnpr2ob  12923  fvprif  12926  xpsfrnel  12927  xpsfeq  12928  2o01f  15487  nninfalllem1  15498  nninfall  15499  nninfsellemqall  15505  nninfomnilem  15508