Theorem ttcuniun 37476

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 4180	. . 3 ⊢ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
2		uniun 4893	. . . . . 6 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
3		ttctr3 37461	. . . . . . 7 ⊢ ∪ TC+ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
4		ttcid 37458	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
5	3, 4	unssi 4192	. . . . . 6 ⊢ (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴) ⊆ TC+ ∪ 𝐴
6	2, 5	eqsstri 3959	. . . . 5 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ ∪ 𝐴
7		ssun3 4181	. . . . 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 5216	. . . 4 ⊢ (Tr (TC+ ∪ 𝐴 ∪ 𝐴) ↔ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
10	8, 9	mpbir 234	. . 3 ⊢ Tr (TC+ ∪ 𝐴 ∪ 𝐴)
11		ttcmin 37462	. . 3 ⊢ ((𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴) ∧ Tr (TC+ ∪ 𝐴 ∪ 𝐴)) → TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
12	1, 10, 11	mp2an 705	. 2 ⊢ TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
13		ttcid 37458	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
14	13	unissi 4879	. . . . 5 ⊢ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
15		ttctr3 37461	. . . . 5 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
16	14, 15	sstri 3922	. . . 4 ⊢ ∪ 𝐴 ⊆ TC+ 𝐴
17		ttcss 37464	. . . 4 ⊢ (∪ 𝐴 ⊆ TC+ 𝐴 → TC+ ∪ 𝐴 ⊆ TC+ 𝐴)
18	16, 17	ax-mp 5	. . 3 ⊢ TC+ ∪ 𝐴 ⊆ TC+ 𝐴
19	18, 13	unssi 4192	. 2 ⊢ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ 𝐴
20	12, 19	eqssi 3929	1 ⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3900   ∪ cun 4152  ∪ cuni 4870  Tr wtr 5215  TC+ cttc 37452

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-ttc 37453

This theorem is used by:  ttcuni  37479