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Theorem hmphindis 21794
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 4326 . . 3 {∅} = {∅, ∅}
2 indislem 20998 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
3 preq2 4405 . . . . . . . 8 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
43, 1syl6eqr 2804 . . . . . . 7 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
52, 4syl5eqr 2800 . . . . . 6 (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
65breq2d 4808 . . . . 5 (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
76biimpac 504 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8 hmph0 21792 . . . 4 (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
97, 8sylib 208 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
109unieqd 4590 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
11 hmphdis.1 . . . . 5 𝑋 = 𝐽
12 0ex 4934 . . . . . . 7 ∅ ∈ V
1312unisn 4595 . . . . . 6 {∅} = ∅
1413eqcomi 2761 . . . . 5 ∅ = {∅}
1510, 11, 143eqtr4g 2811 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
1615preq2d 4411 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
171, 9, 163eqtr4a 2812 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18 hmphen 21782 . . . . . 6 (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
1918adantr 472 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ {∅, 𝐴})
20 necom 2977 . . . . . . . 8 (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
21 fvex 6354 . . . . . . . . 9 ( I ‘𝐴) ∈ V
22 pr2nelem 9009 . . . . . . . . 9 ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
2312, 21, 22mp3an12 1555 . . . . . . . 8 (∅ ≠ ( I ‘𝐴) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
2420, 23sylbi 207 . . . . . . 7 (( I ‘𝐴) ≠ ∅ → {∅, ( I ‘𝐴)} ≈ 2𝑜)
2524adantl 473 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
262, 25syl5eqbrr 4832 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2𝑜)
27 entr 8165 . . . . 5 ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2𝑜) → 𝐽 ≈ 2𝑜)
2819, 26, 27syl2anc 696 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2𝑜)
29 hmphtop1 21776 . . . . . . 7 (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
3029adantr 472 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
3111toptopon 20916 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3230, 31sylib 208 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
33 en2top 20983 . . . . 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 477 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 = {∅, 𝑋})
3717, 36pm2.61dane 3011 1 (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wne 2924  Vcvv 3332  c0 4050  {csn 4313  {cpr 4315   cuni 4580   class class class wbr 4796   I cid 5165  cfv 6041  2𝑜c2o 7715  cen 8110  Topctop 20892  TopOnctopon 20909  chmph 21751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-1o 7721  df-2o 7722  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-top 20893  df-topon 20910  df-cn 21225  df-hmeo 21752  df-hmph 21753
This theorem is referenced by: (None)
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