|   | Mathbox for Chen-Pang He | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > oninhaus | Structured version Visualization version GIF version | ||
| Description: The ordinal Hausdorff spaces are 1o and 2o. (Contributed by Chen-Pang He, 10-Nov-2015.) | 
| Ref | Expression | 
|---|---|
| oninhaus | ⊢ (On ∩ Haus) = {1o, 2o} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | haust1 23361 | . . . . 5 ⊢ (𝑥 ∈ Haus → 𝑥 ∈ Fre) | |
| 2 | 1 | ssriv 3986 | . . . 4 ⊢ Haus ⊆ Fre | 
| 3 | sslin 4242 | . . . 4 ⊢ (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre)) | |
| 4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (On ∩ Haus) ⊆ (On ∩ Fre) | 
| 5 | onint1 36451 | . . 3 ⊢ (On ∩ Fre) = {1o, 2o} | |
| 6 | 4, 5 | sseqtri 4031 | . 2 ⊢ (On ∩ Haus) ⊆ {1o, 2o} | 
| 7 | ssoninhaus 36450 | . 2 ⊢ {1o, 2o} ⊆ (On ∩ Haus) | |
| 8 | 6, 7 | eqssi 3999 | 1 ⊢ (On ∩ Haus) = {1o, 2o} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∩ cin 3949 ⊆ wss 3950 {cpr 4627 Oncon0 6383 1oc1o 8500 2oc2o 8501 Frect1 23316 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-10 2140 ax-11 2156 ax-12 2176 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-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 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-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-1o 8507 df-2o 8508 df-topgen 17489 df-top 22901 df-topon 22918 df-cld 23028 df-t1 23323 df-haus 23324 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |