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Mirrors > Home > MPE Home > Th. List > ord3ex | Structured version Visualization version GIF version |
Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7441. (Contributed by NM, 2-May-2009.) |
Ref | Expression |
---|---|
ord3ex | ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4530 | . 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
2 | pwpr 4794 | . . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) | |
3 | pp0ex 5252 | . . . . 5 ⊢ {∅, {∅}} ∈ V | |
4 | 3 | pwex 5246 | . . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V |
5 | 2, 4 | eqeltrri 2887 | . . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V |
6 | snsspr2 4708 | . . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} | |
7 | unss2 4108 | . . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) |
9 | 5, 8 | ssexi 5190 | . 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V |
10 | 1, 9 | eqeltri 2886 | 1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 {cpr 4527 {ctp 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 |
This theorem is referenced by: (None) |
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