Theorem toponcomb 23247

Description: Biconditional form of toponcom 23246. (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 23246	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐽))
2	1	ex 418	. . 3 ⊢ (𝐾 ∈ Top → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
3	2	adantl 487	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
4		toponcom 23246	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))
5	4	ex 418	. . 3 ⊢ (𝐽 ∈ Top → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
6	5	adantr 486	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
7	3, 6	impbid 215	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ∪ cuni 4866  ‘cfv 6538  Topctop 23211  TopOnctopon 23228

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-topon 23229

This theorem is used by: (None)