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| Description: The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.) | 
| Ref | Expression | 
|---|---|
| ssoninhaus | ⊢ {1o, 2o} ⊆ (On ∩ Haus) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1on 8519 | . . 3 ⊢ 1o ∈ On | |
| 2 | 2on 8521 | . . 3 ⊢ 2o ∈ On | |
| 3 | prssi 4820 | . . 3 ⊢ ((1o ∈ On ∧ 2o ∈ On) → {1o, 2o} ⊆ On) | |
| 4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ {1o, 2o} ⊆ On | 
| 5 | df1o2 8514 | . . . . 5 ⊢ 1o = {∅} | |
| 6 | pw0 4811 | . . . . 5 ⊢ 𝒫 ∅ = {∅} | |
| 7 | 5, 6 | eqtr4i 2767 | . . . 4 ⊢ 1o = 𝒫 ∅ | 
| 8 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
| 9 | dishaus 23391 | . . . . 5 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ Haus) | |
| 10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ 𝒫 ∅ ∈ Haus | 
| 11 | 7, 10 | eqeltri 2836 | . . 3 ⊢ 1o ∈ Haus | 
| 12 | df2o2 8516 | . . . . 5 ⊢ 2o = {∅, {∅}} | |
| 13 | pwpw0 4812 | . . . . 5 ⊢ 𝒫 {∅} = {∅, {∅}} | |
| 14 | 12, 13 | eqtr4i 2767 | . . . 4 ⊢ 2o = 𝒫 {∅} | 
| 15 | p0ex 5383 | . . . . 5 ⊢ {∅} ∈ V | |
| 16 | dishaus 23391 | . . . . 5 ⊢ ({∅} ∈ V → 𝒫 {∅} ∈ Haus) | |
| 17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ 𝒫 {∅} ∈ Haus | 
| 18 | 14, 17 | eqeltri 2836 | . . 3 ⊢ 2o ∈ Haus | 
| 19 | prssi 4820 | . . 3 ⊢ ((1o ∈ Haus ∧ 2o ∈ Haus) → {1o, 2o} ⊆ Haus) | |
| 20 | 11, 18, 19 | mp2an 692 | . 2 ⊢ {1o, 2o} ⊆ Haus | 
| 21 | 4, 20 | ssini 4239 | 1 ⊢ {1o, 2o} ⊆ (On ∩ Haus) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 𝒫 cpw 4599 {csn 4625 {cpr 4627 Oncon0 6383 1oc1o 8500 2oc2o 8501 Hauscha 23317 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-suc 6389 df-1o 8507 df-2o 8508 df-top 22901 df-haus 23324 | 
| This theorem is referenced by: onint1 36451 oninhaus 36452 | 
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