Theorem hmphindis 23705

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 4587	. . 3 ⊢ {∅} = {∅, ∅}
2		indislem 22908	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
3		preq2 4685	. . . . . . . 8 ⊢ (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
4	3, 1	eqtr4di 2783	. . . . . . 7 ⊢ (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
5	2, 4	eqtr3id 2779	. . . . . 6 ⊢ (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
6	5	breq2d 5101	. . . . 5 ⊢ (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
7	6	biimpac 478	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8		hmph0 23703	. . . 4 ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
9	7, 8	sylib 218	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
10	9	unieqd 4870	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → ∪ 𝐽 = ∪ {∅})
11		hmphdis.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
12		0ex 5243	. . . . . . 7 ⊢ ∅ ∈ V
13	12	unisn 4876	. . . . . 6 ⊢ ∪ {∅} = ∅
14	13	eqcomi 2739	. . . . 5 ⊢ ∅ = ∪ {∅}
15	10, 11, 14	3eqtr4g 2790	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
16	15	preq2d 4691	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
17	1, 9, 16	3eqtr4a 2791	. 2 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18		hmphen 23693	. . . . 5 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
19		necom 2979	. . . . . . . 8 ⊢ (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
20		fvex 6830	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ V
21		enpr2 9887	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2o)
22	12, 20, 21	mp3an12 1453	. . . . . . . 8 ⊢ (∅ ≠ ( I ‘𝐴) → {∅, ( I ‘𝐴)} ≈ 2o)
23	19, 22	sylbi 217	. . . . . . 7 ⊢ (( I ‘𝐴) ≠ ∅ → {∅, ( I ‘𝐴)} ≈ 2o)
24	23	adantl 481	. . . . . 6 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, ( I ‘𝐴)} ≈ 2o)
25	2, 24	eqbrtrrid 5125	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2o)
26		entr 8923	. . . . 5 ⊢ ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2o) → 𝐽 ≈ 2o)
27	18, 25, 26	syl2an2r 685	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2o)
28		hmphtop1 23687	. . . . . . 7 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
29	28	adantr 480	. . . . . 6 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
30	11	toptopon 22825	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
31	29, 30	sylib 218	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
32		en2top 22893	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
33	31, 32	syl 17	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
34	27, 33	mpbid 232	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
35	34	simpld 494	. 2 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 = {∅, 𝑋})
36	17, 35	pm2.61dane 3013	1 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ≠ wne 2926  Vcvv 3434  ∅c0 4281  {csn 4574  {cpr 4576  ∪ cuni 4857   class class class wbr 5089   I cid 5508  ‘cfv 6477  2_oc2o 8374   ≈ cen 8861  Topctop 22801  TopOnctopon 22818   ≃ chmph 23662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-card 9824  df-top 22802  df-topon 22819  df-cn 23135  df-hmeo 23663  df-hmph 23664

This theorem is referenced by: (None)