Theorem hmphindis 24149

Description: Homeomorphisms preserve topological indiscreteness. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
hmphdis.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hmphindis	⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})

Proof of Theorem hmphindis

Step	Hyp	Ref	Expression
1		dfsn2 4597	. . 3 ⊢ {∅} = {∅, ∅}
2		indislem 23351	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
3		preq2 4695	. . . . . . . 8 ⊢ (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
4	3, 1	eqtr4di 2816	. . . . . . 7 ⊢ (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
5	2, 4	eqtr3id 2812	. . . . . 6 ⊢ (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
6	5	breq2d 5118	. . . . 5 ⊢ (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
7	6	biimpac 484	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8		hmph0 24147	. . . 4 ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
9	7, 8	sylib 221	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
10	9	unieqd 4883	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → ∪ 𝐽 = ∪ {∅})
11		hmphdis.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
12		0ex 5265	. . . . . . 7 ⊢ ∅ ∈ V
13	12	unisn 4889	. . . . . 6 ⊢ ∪ {∅} = ∅
14	13	eqcomi 2772	. . . . 5 ⊢ ∅ = ∪ {∅}
15	10, 11, 14	3eqtr4g 2823	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
16	15	preq2d 4701	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
17	1, 9, 16	3eqtr4a 2824	. 2 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18		hmphen 24137	. . . . 5 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
19		necom 3012	. . . . . . . 8 ⊢ (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
20		fvex 6924	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ V
21		enpr2 10113	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2o)
22	12, 20, 21	mp3an12 1481	. . . . . . . 8 ⊢ (∅ ≠ ( I ‘𝐴) → {∅, ( I ‘𝐴)} ≈ 2o)
23	19, 22	sylbi 220	. . . . . . 7 ⊢ (( I ‘𝐴) ≠ ∅ → {∅, ( I ‘𝐴)} ≈ 2o)
24	23	adantl 487	. . . . . 6 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, ( I ‘𝐴)} ≈ 2o)
25	2, 24	eqbrtrrid 5145	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2o)
26		entr 9064	. . . . 5 ⊢ ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2o) → 𝐽 ≈ 2o)
27	18, 25, 26	syl2an2r 698	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2o)
28		hmphtop1 24131	. . . . . . 7 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
29	28	adantr 486	. . . . . 6 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
30	11	toptopon 23268	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
31	29, 30	sylib 221	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
32		en2top 23336	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
33	31, 32	syl 18	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
34	27, 33	mpbid 235	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
35	34	simpld 500	. 2 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 = {∅, 𝑋})
36	17, 35	pm2.61dane 3046	1 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  {csn 4588  {cpr 4590  ∪ cuni 4870   class class class wbr 5106   I cid 5550  ‘cfv 6561  2_oc2o 8501   ≈ cen 9001  Topctop 23244  TopOnctopon 23261   ≃ chmph 24106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-1o 8507  df-2o 8508  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-card 10050  df-top 23245  df-topon 23262  df-cn 23578  df-hmeo 24107  df-hmph 24108

This theorem is used by: (None)