Theorem ord3ex 5287

Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7460. (Contributed by NM, 2-May-2009.)

Assertion

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

Proof of Theorem ord3ex

Step	Hyp	Ref	Expression
1		df-tp 4571	. 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2		pwpr 4831	. . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3		pp0ex 5286	. . . . 5 ⊢ {∅, {∅}} ∈ V
4	3	pwex 5280	. . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V
5	2, 4	eqeltrri 2910	. . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6		snsspr2 4747	. . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7		unss2 4156	. . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
8	6, 7	ax-mp 5	. . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
9	5, 8	ssexi 5225	. 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
10	1, 9	eqeltri 2909	1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V

Syntax hints:   ∈ wcel 2110  Vcvv 3494   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4566  {cpr 4568  {ctp 4570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571

This theorem is referenced by: (None)