Theorem toponcomb 13668

Description: Biconditional form of toponcom 13667. (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 13667	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐽))
2	1	ex 115	. . 3 ⊢ (𝐾 ∈ Top → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
3	2	adantl 277	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐽)))
4		toponcom 13667	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))
5	4	ex 115	. . 3 ⊢ (𝐽 ∈ Top → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
6	5	adantr 276	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐾 ∈ (TopOn‘∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐾)))
7	3, 6	impbid 129	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  ∪ cuni 3811  ‘cfv 5218  Topctop 13637  TopOnctopon 13650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-topon 13651

This theorem is referenced by: (None)