Theorem df2o3 6426

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 6413	. 2 ⊢ 2o = suc 1o
2		df-suc 4369	. 2 ⊢ suc 1o = (1o ∪ {1o})
3		df1o2 6425	. . . 4 ⊢ 1o = {∅}
4	3	uneq1i 3285	. . 3 ⊢ (1o ∪ {1o}) = ({∅} ∪ {1o})
5		df-pr 3599	. . 3 ⊢ {∅, 1o} = ({∅} ∪ {1o})
6	4, 5	eqtr4i 2201	. 2 ⊢ (1o ∪ {1o}) = {∅, 1o}
7	1, 2, 6	3eqtri 2202	1 ⊢ 2o = {∅, 1o}

Syntax hints:   = wceq 1353   ∪ cun 3127  ∅c0 3422  {csn 3592  {cpr 3593  suc csuc 4363  1_oc1o 6405  2_oc2o 6406

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-un 3133  df-nul 3423  df-pr 3599  df-suc 4369  df-1o 6412  df-2o 6413

This theorem is referenced by:  df2o2  6427  2oconcl  6435  0lt2o  6437  1lt2o  6438  el2oss1o  6439  en2eqpr  6902  nninfisol  7126  finomni  7133  exmidomniim  7134  exmidomni  7135  ismkvnex  7148  nninfwlpoimlemginf  7169  exmidfodomrlemr  7196  exmidfodomrlemrALT  7197  xp2dju  7209  pw1nel3  7225  sucpw1nel3  7227  unct  12433  2o01f  14517  nninfalllem1  14528  nninfall  14529  nninfsellemqall  14535  nninfomnilem  14538