ssoninhaus - Mathbox for Chen-Pang He

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Theorem ssoninhaus 32447

Description: The ordinal topologies 1_𝑜 and 2_𝑜 are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)

**Assertion**
Ref	Expression
ssoninhaus	⊢ {1_𝑜, 2_𝑜} ⊆ (On ∩ Haus)

**Proof of Theorem ssoninhaus**
Step	Hyp	Ref	Expression
1		1on 7567	. . 3 ⊢ 1_𝑜 ∈ On
2		2on 7568	. . 3 ⊢ 2_𝑜 ∈ On
3		prssi 4353	. . 3 ⊢ ((1_𝑜 ∈ On ∧ 2_𝑜 ∈ On) → {1_𝑜, 2_𝑜} ⊆ On)
4	1, 2, 3	mp2an 708	. 2 ⊢ {1_𝑜, 2_𝑜} ⊆ On
5		df1o2 7572	. . . . 5 ⊢ 1_𝑜 = {∅}
6		pw0 4343	. . . . 5 ⊢ 𝒫 ∅ = {∅}
7	5, 6	eqtr4i 2647	. . . 4 ⊢ 1_𝑜 = 𝒫 ∅
8		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
9		dishaus 21186	. . . . 5 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ Haus)
10	8, 9	ax-mp 5	. . . 4 ⊢ 𝒫 ∅ ∈ Haus
11	7, 10	eqeltri 2697	. . 3 ⊢ 1_𝑜 ∈ Haus
12		df2o2 7574	. . . . 5 ⊢ 2_𝑜 = {∅, {∅}}
13		pwpw0 4344	. . . . 5 ⊢ 𝒫 {∅} = {∅, {∅}}
14	12, 13	eqtr4i 2647	. . . 4 ⊢ 2_𝑜 = 𝒫 {∅}
15		p0ex 4853	. . . . 5 ⊢ {∅} ∈ V
16		dishaus 21186	. . . . 5 ⊢ ({∅} ∈ V → 𝒫 {∅} ∈ Haus)
17	15, 16	ax-mp 5	. . . 4 ⊢ 𝒫 {∅} ∈ Haus
18	14, 17	eqeltri 2697	. . 3 ⊢ 2_𝑜 ∈ Haus
19		prssi 4353	. . 3 ⊢ ((1_𝑜 ∈ Haus ∧ 2_𝑜 ∈ Haus) → {1_𝑜, 2_𝑜} ⊆ Haus)
20	11, 18, 19	mp2an 708	. 2 ⊢ {1_𝑜, 2_𝑜} ⊆ Haus
21	4, 20	ssini 3836	1 ⊢ {1_𝑜, 2_𝑜} ⊆ (On ∩ Haus)

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 {cpr 4179 Oncon0 5723 1_𝑜c1o 7553 2_𝑜c2o 7554 Hauscha 21112

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729 df-1o 7560 df-2o 7561 df-top 20699 df-haus 21119

This theorem is referenced by: onint1 32448 oninhaus 32449

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