Theorem toponcomb 23095

Description: Biconditional form of toponcom 23094. (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 23094	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐽))
2	1	ex 417	. . 3 ⊢ (𝐾 ∈ Top → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
3	2	adantl 486	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
4		toponcom 23094	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))
5	4	ex 417	. . 3 ⊢ (𝐽 ∈ Top → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
6	5	adantr 485	. 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 400   ∈ wcel 2143  ∪ cuni 4872  ‘cfv 6536  Topctop 23059  TopOnctopon 23076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-topon 23077

This theorem is used by: (None)