Theorem topcld 20649

Description: The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
topcld	⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))

Proof of Theorem topcld

Step	Hyp	Ref	Expression
1		difid 3902	. . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅
2		0opn 20534	. . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
3	1, 2	syl5eqel 2692	. . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∖ 𝑋) ∈ 𝐽)
4		ssid 3587	. . 3 ⊢ 𝑋 ⊆ 𝑋
5	3, 4	jctil 558	. 2 ⊢ (𝐽 ∈ Top → (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽))
6		iscld.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
7	6	iscld 20641	. 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽)))
8	5, 7	mpbird 246	1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-cld 20633

This theorem is referenced by:  clsval  20651  riincld  20658  clscld  20661  clstop  20683  cldmre  20692  indiscld  20705  iscon2  21027  cnmpt2pc  22535  rlmbn  22965  ubthlem1  27110  unicls  29277  cmpfiiin  36278  kelac1  36651