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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 7755. (Contributed by NM, 2-May-2009.) |
| Ref | Expression |
|---|---|
| ord3ex | ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-tp 4631 | . 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
| 2 | pwpr 4901 | . . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) | |
| 3 | pp0ex 5386 | . . . . 5 ⊢ {∅, {∅}} ∈ V | |
| 4 | 3 | pwex 5380 | . . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V |
| 5 | 2, 4 | eqeltrri 2838 | . . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V |
| 6 | snsspr2 4815 | . . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} | |
| 7 | unss2 4187 | . . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})) | |
| 8 | 6, 7 | ax-mp 5 | . . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) |
| 9 | 5, 8 | ssexi 5322 | . 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V |
| 10 | 1, 9 | eqeltri 2837 | 1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 {ctp 4630 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 |
| This theorem is referenced by: (None) |
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