Theorem ttcuni 37052

Description: Distribute union of a class through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcuni	⊢ TC+ ∪ 𝐴 = ∪ TC+ 𝐴

Proof of Theorem ttcuni

Step	Hyp	Ref	Expression
1		ttcid 37031	. . . 4 ⊢ 𝐴 ⊆ TC+ 𝐴
2	1	unissi 4880	. . 3 ⊢ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
3		ttctr3 37034	. . . . 5 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
4	3	unissi 4880	. . . 4 ⊢ ∪ ∪ TC+ 𝐴 ⊆ ∪ TC+ 𝐴
5		df-tr 5218	. . . 4 ⊢ (Tr ∪ TC+ 𝐴 ↔ ∪ ∪ TC+ 𝐴 ⊆ ∪ TC+ 𝐴)
6	4, 5	mpbir 234	. . 3 ⊢ Tr ∪ TC+ 𝐴
7		ttcmin 37035	. . 3 ⊢ ((∪ 𝐴 ⊆ ∪ TC+ 𝐴 ∧ Tr ∪ TC+ 𝐴) → TC+ ∪ 𝐴 ⊆ ∪ TC+ 𝐴)
8	2, 6, 7	mp2an 704	. 2 ⊢ TC+ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
9		ttcuniun 37049	. . . . 5 ⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)
10	9	unieqi 4883	. . . 4 ⊢ ∪ TC+ 𝐴 = ∪ (TC+ ∪ 𝐴 ∪ 𝐴)
11		uniun 4894	. . . 4 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
12	10, 11	eqtri 2785	. . 3 ⊢ ∪ TC+ 𝐴 = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
13		ttctr3 37034	. . . 4 ⊢ ∪ TC+ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
14		ttcid 37031	. . . 4 ⊢ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
15	13, 14	unssi 4143	. . 3 ⊢ (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴) ⊆ TC+ ∪ 𝐴
16	12, 15	eqsstri 3982	. 2 ⊢ ∪ TC+ 𝐴 ⊆ TC+ ∪ 𝐴
17	8, 16	eqssi 3952	1 ⊢ TC+ ∪ 𝐴 = ∪ TC+ 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4871  Tr wtr 5217  TC+ cttc 37025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-ttc 37026

This theorem is used by: (None)