Theorem toponcomb 12666

Description: Biconditional form of toponcom 12665. (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 12665	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐽))
2	1	ex 114	. . 3 ⊢ (𝐾 ∈ Top → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
3	2	adantl 275	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
4		toponcom 12665	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))
5	4	ex 114	. . 3 ⊢ (𝐽 ∈ Top → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
6	5	adantr 274	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
7	3, 6	impbid 128	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2136  ∪ cuni 3789  ‘cfv 5188  Topctop 12635  TopOnctopon 12648

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-topon 12649

This theorem is referenced by: (None)