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Theorem hmphindis 21802
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 4334 . . 3 {∅} = {∅, ∅}
2 indislem 21006 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
3 preq2 4413 . . . . . . . 8 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
43, 1syl6eqr 2812 . . . . . . 7 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
52, 4syl5eqr 2808 . . . . . 6 (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
65breq2d 4816 . . . . 5 (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
76biimpac 504 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8 hmph0 21800 . . . 4 (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
97, 8sylib 208 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
109unieqd 4598 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
11 hmphdis.1 . . . . 5 𝑋 = 𝐽
12 0ex 4942 . . . . . . 7 ∅ ∈ V
1312unisn 4603 . . . . . 6 {∅} = ∅
1413eqcomi 2769 . . . . 5 ∅ = {∅}
1510, 11, 143eqtr4g 2819 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
1615preq2d 4419 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
171, 9, 163eqtr4a 2820 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18 hmphen 21790 . . . . . 6 (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
1918adantr 472 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ {∅, 𝐴})
20 necom 2985 . . . . . . . 8 (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
21 fvex 6362 . . . . . . . . 9 ( I ‘𝐴) ∈ V
22 pr2nelem 9017 . . . . . . . . 9 ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2𝑜)
2312, 21, 22mp3an12 1563 . . . . . . . 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 4840 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2𝑜)
27 entr 8173 . . . . 5 ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2𝑜) → 𝐽 ≈ 2𝑜)
2819, 26, 27syl2anc 696 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2𝑜)
29 hmphtop1 21784 . . . . . . 7 (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
3029adantr 472 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
3111toptopon 20924 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3230, 31sylib 208 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
33 en2top 20991 . . . . 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 3019 1 (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  c0 4058  {csn 4321  {cpr 4323   cuni 4588   class class class wbr 4804   I cid 5173  cfv 6049  2𝑜c2o 7723  cen 8118  Topctop 20900  TopOnctopon 20917  chmph 21759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-1o 7729  df-2o 7730  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-top 20901  df-topon 20918  df-cn 21233  df-hmeo 21760  df-hmph 21761
This theorem is referenced by: (None)
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