Theorem topgele 12568

Description: The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)

Assertion

Ref	Expression
topgele	⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋))

Proof of Theorem topgele

Step	Hyp	Ref	Expression
1		topontop 12553	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		0opn 12545	. . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
3	1, 2	syl 14	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
4		toponmax 12564	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5		0ex 4103	. . . 4 ⊢ ∅ ∈ V
6		prssg 3724	. . . 4 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝐽) → ((∅ ∈ 𝐽 ∧ 𝑋 ∈ 𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
7	5, 4, 6	sylancr 411	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((∅ ∈ 𝐽 ∧ 𝑋 ∈ 𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
8	3, 4, 7	mpbi2and 932	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽)
9		toponuni 12554	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
10		eqimss2 3192	. . . 4 ⊢ (𝑋 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝑋)
11	9, 10	syl 14	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ⊆ 𝑋)
12		sspwuni 3944	. . 3 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋)
13	11, 12	sylibr 133	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
14	8, 13	jca 304	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1342   ∈ wcel 2135  Vcvv 2721   ⊆ wss 3111  ∅c0 3404  𝒫 cpw 3553  {cpr 3571  ∪ cuni 3783  ‘cfv 5182  Topctop 12536  TopOnctopon 12549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-top 12537  df-topon 12550

This theorem is referenced by: (None)