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Theorem en2top 20700
Description: If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
en2top (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))

Proof of Theorem en2top
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝐽 ≈ 2𝑜)
2 toponss 20644 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
32ad2ant2rl 784 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑥𝑋)
4 simprl 793 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑋 = ∅)
5 sseq0 3947 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑋 = ∅) → 𝑥 = ∅)
63, 4, 5syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑥 = ∅)
7 velsn 4164 . . . . . . . . . . . . . . . 16 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
86, 7sylibr 224 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑥 ∈ {∅})
98expr 642 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → (𝑥𝐽𝑥 ∈ {∅}))
109ssrdv 3589 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ⊆ {∅})
11 topontop 20641 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
12 0opn 20634 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → ∅ ∈ 𝐽)
1311, 12syl 17 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
1413ad2antrr 761 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → ∅ ∈ 𝐽)
1514snssd 4309 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → {∅} ⊆ 𝐽)
1610, 15eqssd 3600 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 = {∅})
17 0ex 4750 . . . . . . . . . . . . 13 ∅ ∈ V
1817ensn1 7964 . . . . . . . . . . . 12 {∅} ≈ 1𝑜
1916, 18syl6eqbr 4652 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ≈ 1𝑜)
2019olcd 408 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → (𝐽 = ∅ ∨ 𝐽 ≈ 1𝑜))
21 sdom2en01 9068 . . . . . . . . . 10 (𝐽 ≺ 2𝑜 ↔ (𝐽 = ∅ ∨ 𝐽 ≈ 1𝑜))
2220, 21sylibr 224 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ≺ 2𝑜)
23 sdomnen 7928 . . . . . . . . 9 (𝐽 ≺ 2𝑜 → ¬ 𝐽 ≈ 2𝑜)
2422, 23syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → ¬ 𝐽 ≈ 2𝑜)
2524ex 450 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝑋 = ∅ → ¬ 𝐽 ≈ 2𝑜))
2625necon2ad 2805 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝐽 ≈ 2𝑜𝑋 ≠ ∅))
271, 26mpd 15 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝑋 ≠ ∅)
2827necomd 2845 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → ∅ ≠ 𝑋)
2913adantr 481 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → ∅ ∈ 𝐽)
30 toponmax 20643 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3130adantr 481 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝑋𝐽)
32 en2eqpr 8774 . . . . 5 ((𝐽 ≈ 2𝑜 ∧ ∅ ∈ 𝐽𝑋𝐽) → (∅ ≠ 𝑋𝐽 = {∅, 𝑋}))
331, 29, 31, 32syl3anc 1323 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (∅ ≠ 𝑋𝐽 = {∅, 𝑋}))
3428, 33mpd 15 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝐽 = {∅, 𝑋})
3534, 27jca 554 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
36 simprl 793 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝐽 = {∅, 𝑋})
3717a1i 11 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → ∅ ∈ V)
3830adantr 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝑋𝐽)
39 simprr 795 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅)
4039necomd 2845 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → ∅ ≠ 𝑋)
41 pr2nelem 8771 . . . 4 ((∅ ∈ V ∧ 𝑋𝐽 ∧ ∅ ≠ 𝑋) → {∅, 𝑋} ≈ 2𝑜)
4237, 38, 40, 41syl3anc 1323 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → {∅, 𝑋} ≈ 2𝑜)
4336, 42eqbrtrd 4635 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝐽 ≈ 2𝑜)
4435, 43impbida 876 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  wss 3555  c0 3891  {csn 4148  {cpr 4150   class class class wbr 4613  cfv 5847  1𝑜c1o 7498  2𝑜c2o 7499  cen 7896  csdm 7898  Topctop 20617  TopOnctopon 20618
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-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-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-om 7013  df-1o 7505  df-2o 7506  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-top 20621  df-topon 20623
This theorem is referenced by:  hmphindis  21510
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