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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 7454 | . . 3 ⊢ 1𝑜 ∈ On | |
2 | 2on 7455 | . . 3 ⊢ 2𝑜 ∈ On | |
3 | prssi 4293 | . . 3 ⊢ ((1𝑜 ∈ On ∧ 2𝑜 ∈ On) → {1𝑜, 2𝑜} ⊆ On) | |
4 | 1, 2, 3 | mp2an 704 | . 2 ⊢ {1𝑜, 2𝑜} ⊆ On |
5 | df1o2 7459 | . . . . 5 ⊢ 1𝑜 = {∅} | |
6 | pw0 4283 | . . . . 5 ⊢ 𝒫 ∅ = {∅} | |
7 | 5, 6 | eqtr4i 2635 | . . . 4 ⊢ 1𝑜 = 𝒫 ∅ |
8 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
9 | dishaus 20996 | . . . . 5 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ Haus) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ 𝒫 ∅ ∈ Haus |
11 | 7, 10 | eqeltri 2684 | . . 3 ⊢ 1𝑜 ∈ Haus |
12 | df2o2 7461 | . . . . 5 ⊢ 2𝑜 = {∅, {∅}} | |
13 | pwpw0 4284 | . . . . 5 ⊢ 𝒫 {∅} = {∅, {∅}} | |
14 | 12, 13 | eqtr4i 2635 | . . . 4 ⊢ 2𝑜 = 𝒫 {∅} |
15 | p0ex 4779 | . . . . 5 ⊢ {∅} ∈ V | |
16 | dishaus 20996 | . . . . 5 ⊢ ({∅} ∈ V → 𝒫 {∅} ∈ Haus) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ 𝒫 {∅} ∈ Haus |
18 | 14, 17 | eqeltri 2684 | . . 3 ⊢ 2𝑜 ∈ Haus |
19 | prssi 4293 | . . 3 ⊢ ((1𝑜 ∈ Haus ∧ 2𝑜 ∈ Haus) → {1𝑜, 2𝑜} ⊆ Haus) | |
20 | 11, 18, 19 | mp2an 704 | . 2 ⊢ {1𝑜, 2𝑜} ⊆ Haus |
21 | 4, 20 | ssini 3798 | 1 ⊢ {1𝑜, 2𝑜} ⊆ (On ∩ Haus) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 Oncon0 5640 1𝑜c1o 7440 2𝑜c2o 7441 Hauscha 20922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-suc 5646 df-1o 7447 df-2o 7448 df-top 20521 df-haus 20929 |
This theorem is referenced by: onint1 31618 oninhaus 31619 |
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