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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssoninhaus | Structured version Visualization version GIF version |
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 8198 | . . 3 ⊢ 1o ∈ On | |
2 | 2on 8199 | . . 3 ⊢ 2o ∈ On | |
3 | prssi 4724 | . . 3 ⊢ ((1o ∈ On ∧ 2o ∈ On) → {1o, 2o} ⊆ On) | |
4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ {1o, 2o} ⊆ On |
5 | df1o2 8203 | . . . . 5 ⊢ 1o = {∅} | |
6 | pw0 4715 | . . . . 5 ⊢ 𝒫 ∅ = {∅} | |
7 | 5, 6 | eqtr4i 2765 | . . . 4 ⊢ 1o = 𝒫 ∅ |
8 | 0ex 5189 | . . . . 5 ⊢ ∅ ∈ V | |
9 | dishaus 22251 | . . . . 5 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ Haus) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ 𝒫 ∅ ∈ Haus |
11 | 7, 10 | eqeltri 2830 | . . 3 ⊢ 1o ∈ Haus |
12 | df2o2 8207 | . . . . 5 ⊢ 2o = {∅, {∅}} | |
13 | pwpw0 4716 | . . . . 5 ⊢ 𝒫 {∅} = {∅, {∅}} | |
14 | 12, 13 | eqtr4i 2765 | . . . 4 ⊢ 2o = 𝒫 {∅} |
15 | p0ex 5266 | . . . . 5 ⊢ {∅} ∈ V | |
16 | dishaus 22251 | . . . . 5 ⊢ ({∅} ∈ V → 𝒫 {∅} ∈ Haus) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ 𝒫 {∅} ∈ Haus |
18 | 14, 17 | eqeltri 2830 | . . 3 ⊢ 2o ∈ Haus |
19 | prssi 4724 | . . 3 ⊢ ((1o ∈ Haus ∧ 2o ∈ Haus) → {1o, 2o} ⊆ Haus) | |
20 | 11, 18, 19 | mp2an 692 | . 2 ⊢ {1o, 2o} ⊆ Haus |
21 | 4, 20 | ssini 4136 | 1 ⊢ {1o, 2o} ⊆ (On ∩ Haus) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3401 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 {csn 4531 {cpr 4533 Oncon0 6202 1oc1o 8184 2oc2o 8185 Hauscha 22177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-tr 5151 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-ord 6205 df-on 6206 df-suc 6208 df-1o 8191 df-2o 8192 df-top 21763 df-haus 22184 |
This theorem is referenced by: onint1 34332 oninhaus 34333 |
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