Theorem toponcomb 21986

Description: Biconditional form of toponcom 21985. (Contributed by BJ, 5-Dec-2021.)

Assertion

Ref	Expression
toponcomb	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))

Proof of Theorem toponcomb

Step	Hyp	Ref	Expression
1		toponcom 21985	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐽))
2	1	ex 412	. . 3 ⊢ (𝐾 ∈ Top → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
3	2	adantl 481	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
4		toponcom 21985	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))
5	4	ex 412	. . 3 ⊢ (𝐽 ∈ Top → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
6	5	adantr 480	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
7	3, 6	impbid 211	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  TopOnctopon 21967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-topon 21968

This theorem is referenced by: (None)