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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 21679 | . . . . 5 ⊢ (𝑥 ∈ Haus → 𝑥 ∈ Fre) | |
2 | 1 | ssriv 3855 | . . . 4 ⊢ Haus ⊆ Fre |
3 | sslin 4092 | . . . 4 ⊢ (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (On ∩ Haus) ⊆ (On ∩ Fre) |
5 | onint1 33354 | . . 3 ⊢ (On ∩ Fre) = {1o, 2o} | |
6 | 4, 5 | sseqtri 3886 | . 2 ⊢ (On ∩ Haus) ⊆ {1o, 2o} |
7 | ssoninhaus 33353 | . 2 ⊢ {1o, 2o} ⊆ (On ∩ Haus) | |
8 | 6, 7 | eqssi 3867 | 1 ⊢ (On ∩ Haus) = {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 ∩ cin 3821 ⊆ wss 3822 {cpr 4437 Oncon0 6026 1oc1o 7896 2oc2o 7897 Frect1 21634 Hauscha 21635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-ord 6029 df-on 6030 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-fv 6193 df-1o 7903 df-2o 7904 df-topgen 16571 df-top 21221 df-topon 21238 df-cld 21346 df-t1 21641 df-haus 21642 |
This theorem is referenced by: (None) |
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