Theorem df2o3 6320

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 6307	. 2 ⊢ 2o = suc 1o
2		df-suc 4288	. 2 ⊢ suc 1o = (1o ∪ {1o})
3		df1o2 6319	. . . 4 ⊢ 1o = {∅}
4	3	uneq1i 3221	. . 3 ⊢ (1o ∪ {1o}) = ({∅} ∪ {1o})
5		df-pr 3529	. . 3 ⊢ {∅, 1o} = ({∅} ∪ {1o})
6	4, 5	eqtr4i 2161	. 2 ⊢ (1o ∪ {1o}) = {∅, 1o}
7	1, 2, 6	3eqtri 2162	1 ⊢ 2o = {∅, 1o}

Syntax hints:   = wceq 1331   ∪ cun 3064  ∅c0 3358  {csn 3522  {cpr 3523  suc csuc 4282  1_oc1o 6299  2_oc2o 6300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-nul 3359  df-pr 3529  df-suc 4288  df-1o 6306  df-2o 6307

This theorem is referenced by:  df2o2  6321  2oconcl  6329  0lt2o  6331  1lt2o  6332  en2eqpr  6794  finomni  7005  exmidomniim  7006  exmidomni  7007  ismkvnex  7022  exmidfodomrlemr  7051  exmidfodomrlemrALT  7052  xp2dju  7064  unct  11943  el2oss1o  13177  nninfalllem1  13192  nninfall  13193  nninfsellemqall  13200  nninfomnilem  13203  isomninnlem  13214