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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 23376 | . . . . 5 ⊢ (𝑥 ∈ Haus → 𝑥 ∈ Fre) | |
2 | 1 | ssriv 3999 | . . . 4 ⊢ Haus ⊆ Fre |
3 | sslin 4251 | . . . 4 ⊢ (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (On ∩ Haus) ⊆ (On ∩ Fre) |
5 | onint1 36432 | . . 3 ⊢ (On ∩ Fre) = {1o, 2o} | |
6 | 4, 5 | sseqtri 4032 | . 2 ⊢ (On ∩ Haus) ⊆ {1o, 2o} |
7 | ssoninhaus 36431 | . 2 ⊢ {1o, 2o} ⊆ (On ∩ Haus) | |
8 | 6, 7 | eqssi 4012 | 1 ⊢ (On ∩ Haus) = {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3962 ⊆ wss 3963 {cpr 4633 Oncon0 6386 1oc1o 8498 2oc2o 8499 Frect1 23331 Hauscha 23332 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-1o 8505 df-2o 8506 df-topgen 17490 df-top 22916 df-topon 22933 df-cld 23043 df-t1 23338 df-haus 23339 |
This theorem is referenced by: (None) |
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