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Theorem hmphindis 21510
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
StepHypRef Expression
1 dfsn2 4161 . . 3 {∅} = {∅, ∅}
2 indislem 20714 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
3 preq2 4239 . . . . . . . 8 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
43, 1syl6eqr 2673 . . . . . . 7 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
52, 4syl5eqr 2669 . . . . . 6 (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
65breq2d 4625 . . . . 5 (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
76biimpac 503 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8 hmph0 21508 . . . 4 (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
97, 8sylib 208 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
109unieqd 4412 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
11 hmphdis.1 . . . . 5 𝑋 = 𝐽
12 0ex 4750 . . . . . . 7 ∅ ∈ V
1312unisn 4417 . . . . . 6 {∅} = ∅
1413eqcomi 2630 . . . . 5 ∅ = {∅}
1510, 11, 143eqtr4g 2680 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
1615preq2d 4245 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
171, 9, 163eqtr4a 2681 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18 hmphen 21498 . . . . . 6 (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
1918adantr 481 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ {∅, 𝐴})
20 necom 2843 . . . . . . . 8 (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
21 fvex 6158 . . . . . . . . 9 ( I ‘𝐴) ∈ V
22 pr2nelem 8771 . . . . . . . . 9 ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
2312, 21, 22mp3an12 1411 . . . . . . . 8 (∅ ≠ ( I ‘𝐴) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
2420, 23sylbi 207 . . . . . . 7 (( I ‘𝐴) ≠ ∅ → {∅, ( I ‘𝐴)} ≈ 2𝑜)
2524adantl 482 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
262, 25syl5eqbrr 4649 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2𝑜)
27 entr 7952 . . . . 5 ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2𝑜) → 𝐽 ≈ 2𝑜)
2819, 26, 27syl2anc 692 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2𝑜)
29 hmphtop1 21492 . . . . . . 7 (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
3029adantr 481 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
3111toptopon 20648 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3230, 31sylib 208 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
33 en2top 20700 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
3432, 33syl 17 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
3528, 34mpbid 222 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
3635simpld 475 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 = {∅, 𝑋})
3717, 36pm2.61dane 2877 1 (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  c0 3891  {csn 4148  {cpr 4150   cuni 4402   class class class wbr 4613   I cid 4984  cfv 5847  2𝑜c2o 7499  cen 7896  Topctop 20617  TopOnctopon 20618  chmph 21467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-1o 7505  df-2o 7506  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-top 20621  df-topon 20623  df-cn 20941  df-hmeo 21468  df-hmph 21469
This theorem is referenced by: (None)
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