MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ord3ex Structured version   Visualization version   GIF version

Theorem ord3ex 5005
Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7115. (Contributed by NM, 2-May-2009.)
Assertion
Ref Expression
ord3ex {∅, {∅}, {∅, {∅}}} ∈ V

Proof of Theorem ord3ex
StepHypRef Expression
1 df-tp 4326 . 2 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2 pwpr 4582 . . . 4 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3 pp0ex 5004 . . . . 5 {∅, {∅}} ∈ V
43pwex 4997 . . . 4 𝒫 {∅, {∅}} ∈ V
52, 4eqeltrri 2836 . . 3 ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6 snsspr2 4491 . . . 4 {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7 unss2 3927 . . . 4 ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
86, 7ax-mp 5 . . 3 ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
95, 8ssexi 4955 . 2 ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
101, 9eqeltri 2835 1 {∅, {∅}, {∅, {∅}}} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2139  Vcvv 3340  cun 3713  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  {cpr 4323  {ctp 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator