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Theorem en2top 21010
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 471 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝐽 ≈ 2𝑜)
2 toponss 20952 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
32ad2ant2rl 743 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑥𝑋)
4 simprl 754 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑋 = ∅)
5 sseq0 4120 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑋 = ∅) → 𝑥 = ∅)
63, 4, 5syl2anc 573 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑥 = ∅)
7 velsn 4333 . . . . . . . . . . . . . . . 16 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
86, 7sylibr 224 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ (𝑋 = ∅ ∧ 𝑥𝐽)) → 𝑥 ∈ {∅})
98expr 444 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → (𝑥𝐽𝑥 ∈ {∅}))
109ssrdv 3758 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ⊆ {∅})
11 topontop 20938 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
12 0opn 20929 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → ∅ ∈ 𝐽)
1311, 12syl 17 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
1413ad2antrr 705 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → ∅ ∈ 𝐽)
1514snssd 4476 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → {∅} ⊆ 𝐽)
1610, 15eqssd 3769 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 = {∅})
17 0ex 4925 . . . . . . . . . . . . 13 ∅ ∈ V
1817ensn1 8177 . . . . . . . . . . . 12 {∅} ≈ 1𝑜
1916, 18syl6eqbr 4826 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ≈ 1𝑜)
2019olcd 863 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → (𝐽 = ∅ ∨ 𝐽 ≈ 1𝑜))
21 sdom2en01 9330 . . . . . . . . . 10 (𝐽 ≺ 2𝑜 ↔ (𝐽 = ∅ ∨ 𝐽 ≈ 1𝑜))
2220, 21sylibr 224 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → 𝐽 ≺ 2𝑜)
23 sdomnen 8142 . . . . . . . . 9 (𝐽 ≺ 2𝑜 → ¬ 𝐽 ≈ 2𝑜)
2422, 23syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) ∧ 𝑋 = ∅) → ¬ 𝐽 ≈ 2𝑜)
2524ex 397 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝑋 = ∅ → ¬ 𝐽 ≈ 2𝑜))
2625necon2ad 2958 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝐽 ≈ 2𝑜𝑋 ≠ ∅))
271, 26mpd 15 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝑋 ≠ ∅)
2827necomd 2998 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → ∅ ≠ 𝑋)
2913adantr 466 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → ∅ ∈ 𝐽)
30 toponmax 20951 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3130adantr 466 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝑋𝐽)
32 en2eqpr 9034 . . . . 5 ((𝐽 ≈ 2𝑜 ∧ ∅ ∈ 𝐽𝑋𝐽) → (∅ ≠ 𝑋𝐽 = {∅, 𝑋}))
331, 29, 31, 32syl3anc 1476 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (∅ ≠ 𝑋𝐽 = {∅, 𝑋}))
3428, 33mpd 15 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → 𝐽 = {∅, 𝑋})
3534, 27jca 501 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ≈ 2𝑜) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
36 simprl 754 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝐽 = {∅, 𝑋})
3717a1i 11 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → ∅ ∈ V)
3830adantr 466 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝑋𝐽)
39 simprr 756 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅)
4039necomd 2998 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → ∅ ≠ 𝑋)
41 pr2nelem 9031 . . . 4 ((∅ ∈ V ∧ 𝑋𝐽 ∧ ∅ ≠ 𝑋) → {∅, 𝑋} ≈ 2𝑜)
4237, 38, 40, 41syl3anc 1476 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → {∅, 𝑋} ≈ 2𝑜)
4336, 42eqbrtrd 4809 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)) → 𝐽 ≈ 2𝑜)
4435, 43impbida 802 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  wss 3723  c0 4063  {csn 4317  {cpr 4319   class class class wbr 4787  cfv 6030  1𝑜c1o 7710  2𝑜c2o 7711  cen 8110  csdm 8112  Topctop 20918  TopOnctopon 20935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-om 7217  df-1o 7717  df-2o 7718  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-top 20919  df-topon 20936
This theorem is referenced by:  hmphindis  21821
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