Theorem en2top 20789

Description: If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
en2top	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))

Proof of Theorem en2top

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝐽 ≈ 2𝑜)
2		toponss 20731	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
3	2	ad2ant2rl 785	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥 ∈ 𝐽)) → 𝑥 ⊆ 𝑋)
4		simprl 794	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥 ∈ 𝐽)) → 𝑋 = ∅)
5		sseq0 3975	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑋 = ∅) → 𝑥 = ∅)
6	3, 4, 5	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥 ∈ 𝐽)) → 𝑥 = ∅)
7		velsn 4193	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
8	6, 7	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥 ∈ 𝐽)) → 𝑥 ∈ {∅})
9	8	expr 643	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → (𝑥 ∈ 𝐽 → 𝑥 ∈ {∅}))
10	9	ssrdv 3609	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ⊆ {∅})
11		topontop 20718	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
12		0opn 20709	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
13	11, 12	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
14	13	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → ∅ ∈ 𝐽)
15	14	snssd 4340	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → {∅} ⊆ 𝐽)
16	10, 15	eqssd 3620	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 = {∅})
17		0ex 4790	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
18	17	ensn1 8020	. . . . . . . . . . . 12 ⊢ {∅} ≈ 1𝑜
19	16, 18	syl6eqbr 4692	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ≈ 1𝑜)
20	19	olcd 408	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → (𝐽 = ∅ ∨ 𝐽 ≈ 1𝑜))
21		sdom2en01 9124	. . . . . . . . . 10 ⊢ (𝐽 ≺ 2𝑜 ↔ (𝐽 = ∅ ∨ 𝐽 ≈ 1𝑜))
22	20, 21	sylibr 224	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ≺ 2𝑜)
23		sdomnen 7984	. . . . . . . . 9 ⊢ (𝐽 ≺ 2𝑜 → ¬ 𝐽 ≈ 2𝑜)
24	22, 23	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → ¬ 𝐽 ≈ 2𝑜)
25	24	ex 450	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝑋 = ∅ → ¬ 𝐽 ≈ 2𝑜))
26	25	necon2ad 2809	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝐽 ≈ 2𝑜 → 𝑋 ≠ ∅))
27	1, 26	mpd 15	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝑋 ≠ ∅)
28	27	necomd 2849	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → ∅ ≠ 𝑋)
29	13	adantr 481	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → ∅ ∈ 𝐽)
30		toponmax 20730	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
31	30	adantr 481	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝑋 ∈ 𝐽)
32		en2eqpr 8830	. . . . 5 ⊢ ((𝐽 ≈ 2𝑜 ∧ ∅ ∈ 𝐽 ∧ 𝑋 ∈ 𝐽) → (∅ ≠ 𝑋 → 𝐽 = {∅, 𝑋}))
33	1, 29, 31, 32	syl3anc 1326	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (∅ ≠ 𝑋 → 𝐽 = {∅, 𝑋}))
34	28, 33	mpd 15	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝐽 = {∅, 𝑋})
35	34, 27	jca 554	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
36		simprl 794	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝐽 = {∅, 𝑋})
37	17	a1i 11	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → ∅ ∈ V)
38	30	adantr 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝑋 ∈ 𝐽)
39		simprr 796	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅)
40	39	necomd 2849	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → ∅ ≠ 𝑋)
41		pr2nelem 8827	. . . 4 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝐽 ∧ ∅ ≠ 𝑋) → {∅, 𝑋} ≈ 2𝑜)
42	37, 38, 40, 41	syl3anc 1326	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → {∅, 𝑋} ≈ 2𝑜)
43	36, 42	eqbrtrd 4675	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝐽 ≈ 2𝑜)
44	35, 43	impbida 877	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179   class class class wbr 4653  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554   ≈ cen 7952   ≺ csdm 7954  Topctop 20698  TopOnctopon 20715

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-reu 2919  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-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-top 20699  df-topon 20716

This theorem is referenced by:  hmphindis  21600