Theorem hmphindis 21410

Description: Homeomorphisms preserve topological indiscretion. (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 4138	. . 3 ⊢ {∅} = {∅, ∅}
2		indislem 20614	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
3		preq2 4213	. . . . . . . 8 ⊢ (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
4	3, 1	syl6eqr 2662	. . . . . . 7 ⊢ (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
5	2, 4	syl5eqr 2658	. . . . . 6 ⊢ (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
6	5	breq2d 4595	. . . . 5 ⊢ (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
7	6	biimpac 502	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8		hmph0 21408	. . . 4 ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
9	7, 8	sylib 207	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
10	9	unieqd 4382	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → ∪ 𝐽 = ∪ {∅})
11		hmphdis.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
12		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
13	12	unisn 4387	. . . . . 6 ⊢ ∪ {∅} = ∅
14	13	eqcomi 2619	. . . . 5 ⊢ ∅ = ∪ {∅}
15	10, 11, 14	3eqtr4g 2669	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
16	15	preq2d 4219	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
17	1, 9, 16	3eqtr4a 2670	. 2 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18		hmphen 21398	. . . . . 6 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
19	18	adantr 480	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ {∅, 𝐴})
20		necom 2835	. . . . . . . 8 ⊢ (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
21		fvex 6113	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ V
22		pr2nelem 8710	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
23	12, 21, 22	mp3an12 1406	. . . . . . . 8 ⊢ (∅ ≠ ( I ‘𝐴) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
24	20, 23	sylbi 206	. . . . . . 7 ⊢ (( I ‘𝐴) ≠ ∅ → {∅, ( I ‘𝐴)} ≈ 2𝑜)
25	24	adantl 481	. . . . . 6 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
26	2, 25	syl5eqbrr 4619	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2𝑜)
27		entr 7894	. . . . 5 ⊢ ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2𝑜) → 𝐽 ≈ 2𝑜)
28	19, 26, 27	syl2anc 691	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2𝑜)
29		hmphtop1 21392	. . . . . . 7 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
30	29	adantr 480	. . . . . 6 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
31	11	toptopon 20548	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
32	30, 31	sylib 207	. . . . 5 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
33		en2top 20600	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
34	32, 33	syl 17	. . . 4 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
35	28, 34	mpbid 221	. . 3 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
36	35	simpld 474	. 2 ⊢ ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 = {∅, 𝑋})
37	17, 36	pm2.61dane 2869	1 ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127  ∪ cuni 4372   class class class wbr 4583   I cid 4948  ‘cfv 5804  2_𝑜c2o 7441   ≈ cen 7838  Topctop 20517  TopOnctopon 20518   ≃ chmph 21367

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-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-top 20521  df-topon 20523  df-cn 20841  df-hmeo 21368  df-hmph 21369

This theorem is referenced by: (None)