Theorem ttcuniun 36680

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 4110	. . 3 ⊢ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
2		uniun 4863	. . . . . 6 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
3		ttctr3 36665	. . . . . . 7 ⊢ ∪ TC+ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
4		ttcid 36662	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
5	3, 4	unssi 4122	. . . . . 6 ⊢ (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴) ⊆ TC+ ∪ 𝐴
6	2, 5	eqsstri 3963	. . . . 5 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ ∪ 𝐴
7		ssun3 4111	. . . . 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 5182	. . . 4 ⊢ (Tr (TC+ ∪ 𝐴 ∪ 𝐴) ↔ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
10	8, 9	mpbir 231	. . 3 ⊢ Tr (TC+ ∪ 𝐴 ∪ 𝐴)
11		ttcmin 36666	. . 3 ⊢ ((𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴) ∧ Tr (TC+ ∪ 𝐴 ∪ 𝐴)) → TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
12	1, 10, 11	mp2an 693	. 2 ⊢ TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
13		ttcid 36662	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
14	13	unissi 4849	. . . . 5 ⊢ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
15		ttctr3 36665	. . . . 5 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
16	14, 15	sstri 3926	. . . 4 ⊢ ∪ 𝐴 ⊆ TC+ 𝐴
17		ttcss 36668	. . . 4 ⊢ (∪ 𝐴 ⊆ TC+ 𝐴 → TC+ ∪ 𝐴 ⊆ TC+ 𝐴)
18	16, 17	ax-mp 5	. . 3 ⊢ TC+ ∪ 𝐴 ⊆ TC+ 𝐴
19	18, 13	unssi 4122	. 2 ⊢ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ 𝐴
20	12, 19	eqssi 3933	1 ⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)

Syntax hints:   = wceq 1542   ∪ cun 3883   ⊆ wss 3885  ∪ cuni 4840  Tr wtr 5181  TC+ cttc 36656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-ttc 36657

This theorem is referenced by:  ttcuni  36683