Theorem ttcuniun 36875

Description: Relationship between TC+ 𝐴 and TC+ ∪ 𝐴: we can decompose TC+ 𝐴 into the elements of TC+ ∪ 𝐴 plus the elements of 𝐴 itself. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcuniun	⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)

Proof of Theorem ttcuniun

Step	Hyp	Ref	Expression
1		ssun2 4132	. . 3 ⊢ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
2		uniun 4889	. . . . . 6 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
3		ttctr3 36860	. . . . . . 7 ⊢ ∪ TC+ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
4		ttcid 36857	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
5	3, 4	unssi 4144	. . . . . 6 ⊢ (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴) ⊆ TC+ ∪ 𝐴
6	2, 5	eqsstri 3983	. . . . 5 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ ∪ 𝐴
7		ssun3 4133	. . . . 5 ⊢ (∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ ∪ 𝐴 → ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
9		df-tr 5209	. . . 4 ⊢ (Tr (TC+ ∪ 𝐴 ∪ 𝐴) ↔ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
10	8, 9	mpbir 233	. . 3 ⊢ Tr (TC+ ∪ 𝐴 ∪ 𝐴)
11		ttcmin 36861	. . 3 ⊢ ((𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴) ∧ Tr (TC+ ∪ 𝐴 ∪ 𝐴)) → TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
12	1, 10, 11	mp2an 702	. 2 ⊢ TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
13		ttcid 36857	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
14	13	unissi 4875	. . . . 5 ⊢ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
15		ttctr3 36860	. . . . 5 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
16	14, 15	sstri 3946	. . . 4 ⊢ ∪ 𝐴 ⊆ TC+ 𝐴
17		ttcss 36863	. . . 4 ⊢ (∪ 𝐴 ⊆ TC+ 𝐴 → TC+ ∪ 𝐴 ⊆ TC+ 𝐴)
18	16, 17	ax-mp 5	. . 3 ⊢ TC+ ∪ 𝐴 ⊆ TC+ 𝐴
19	18, 13	unssi 4144	. 2 ⊢ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ 𝐴
20	12, 19	eqssi 3953	1 ⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)

Syntax hints:   = wceq 1561   ∪ cun 3903   ⊆ wss 3905  ∪ cuni 4866  Tr wtr 5208  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-ttc 36852

This theorem is referenced by:  ttcuni  36878