Theorem ord3ex 5359

Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7733. (Contributed by NM, 2-May-2009.)

Assertion

Ref	Expression
ord3ex	⊢ {∅, {∅}, {∅, {∅}}} ∈ V

Proof of Theorem ord3ex

Step	Hyp	Ref	Expression
1		df-tp 4599	. 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2		pwpr 4870	. . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3		pp0ex 5358	. . . . 5 ⊢ {∅, {∅}} ∈ V
4	3	pwex 5352	. . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V
5	2, 4	eqeltrri 2866	. . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6		snsspr2 4785	. . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7		unss2 4148	. . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
8	6, 7	ax-mp 5	. . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
9	5, 8	ssexi 5293	. 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
10	1, 9	eqeltri 2865	1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V

Syntax hints:   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  {cpr 4596  {ctp 4598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599

This theorem is referenced by: (None)