Theorem ttcuni 37223

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 37202	. . . 4 ⊢ 𝐴 ⊆ TC+ 𝐴
2	1	unissi 4875	. . 3 ⊢ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
3		ttctr3 37205	. . . . 5 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
4	3	unissi 4875	. . . 4 ⊢ ∪ ∪ TC+ 𝐴 ⊆ ∪ TC+ 𝐴
5		df-tr 5212	. . . 4 ⊢ (Tr ∪ TC+ 𝐴 ↔ ∪ ∪ TC+ 𝐴 ⊆ ∪ TC+ 𝐴)
6	4, 5	mpbir 234	. . 3 ⊢ Tr ∪ TC+ 𝐴
7		ttcmin 37206	. . 3 ⊢ ((∪ 𝐴 ⊆ ∪ TC+ 𝐴 ∧ Tr ∪ TC+ 𝐴) → TC+ ∪ 𝐴 ⊆ ∪ TC+ 𝐴)
8	2, 6, 7	mp2an 705	. 2 ⊢ TC+ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
9		ttcuniun 37220	. . . . 5 ⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)
10	9	unieqi 4878	. . . 4 ⊢ ∪ TC+ 𝐴 = ∪ (TC+ ∪ 𝐴 ∪ 𝐴)
11		uniun 4889	. . . 4 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
12	10, 11	eqtri 2783	. . 3 ⊢ ∪ TC+ 𝐴 = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
13		ttctr3 37205	. . . 4 ⊢ ∪ TC+ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
14		ttcid 37202	. . . 4 ⊢ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
15	13, 14	unssi 4136	. . 3 ⊢ (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴) ⊆ TC+ ∪ 𝐴
16	12, 15	eqsstri 3976	. 2 ⊢ ∪ TC+ 𝐴 ⊆ TC+ ∪ 𝐴
17	8, 16	eqssi 3946	1 ⊢ TC+ ∪ 𝐴 = ∪ TC+ 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3896   ⊆ wss 3898  ∪ cuni 4866  Tr wtr 5211  TC+ cttc 37196

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 37197

This theorem is used by: (None)