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Theorem hmphindis 23821
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
StepHypRef Expression
1 dfsn2 4644 . . 3 {∅} = {∅, ∅}
2 indislem 23023 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
3 preq2 4739 . . . . . . . 8 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
43, 1eqtr4di 2793 . . . . . . 7 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
52, 4eqtr3id 2789 . . . . . 6 (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
65breq2d 5160 . . . . 5 (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
76biimpac 478 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8 hmph0 23819 . . . 4 (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
97, 8sylib 218 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
109unieqd 4925 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
11 hmphdis.1 . . . . 5 𝑋 = 𝐽
12 0ex 5313 . . . . . . 7 ∅ ∈ V
1312unisn 4931 . . . . . 6 {∅} = ∅
1413eqcomi 2744 . . . . 5 ∅ = {∅}
1510, 11, 143eqtr4g 2800 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
1615preq2d 4745 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
171, 9, 163eqtr4a 2801 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18 hmphen 23809 . . . . 5 (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
19 necom 2992 . . . . . . . 8 (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
20 fvex 6920 . . . . . . . . 9 ( I ‘𝐴) ∈ V
21 enpr2 10040 . . . . . . . . 9 ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2o)
2212, 20, 21mp3an12 1450 . . . . . . . 8 (∅ ≠ ( I ‘𝐴) → {∅, ( I ‘𝐴)} ≈ 2o)
2319, 22sylbi 217 . . . . . . 7 (( I ‘𝐴) ≠ ∅ → {∅, ( I ‘𝐴)} ≈ 2o)
2423adantl 481 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, ( I ‘𝐴)} ≈ 2o)
252, 24eqbrtrrid 5184 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2o)
26 entr 9045 . . . . 5 ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2o) → 𝐽 ≈ 2o)
2718, 25, 26syl2an2r 685 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2o)
28 hmphtop1 23803 . . . . . . 7 (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
2928adantr 480 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
3011toptopon 22939 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3129, 30sylib 218 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
32 en2top 23008 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
3331, 32syl 17 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
3427, 33mpbid 232 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
3534simpld 494 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 = {∅, 𝑋})
3617, 35pm2.61dane 3027 1 (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  c0 4339  {csn 4631  {cpr 4633   cuni 4912   class class class wbr 5148   I cid 5582  cfv 6563  2oc2o 8499  cen 8981  Topctop 22915  TopOnctopon 22932  chmph 23778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-top 22916  df-topon 22933  df-cn 23251  df-hmeo 23779  df-hmph 23780
This theorem is referenced by: (None)
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