Theorem ttcuniun 36965

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 4131	. . 3 ⊢ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
2		uniun 4894	. . . . . 6 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) = (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴)
3		ttctr3 36950	. . . . . . 7 ⊢ ∪ TC+ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
4		ttcid 36947	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ TC+ ∪ 𝐴
5	3, 4	unssi 4143	. . . . . 6 ⊢ (∪ TC+ ∪ 𝐴 ∪ ∪ 𝐴) ⊆ TC+ ∪ 𝐴
6	2, 5	eqsstri 3982	. . . . 5 ⊢ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ ∪ 𝐴
7		ssun3 4132	. . . . 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 5218	. . . 4 ⊢ (Tr (TC+ ∪ 𝐴 ∪ 𝐴) ↔ ∪ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
10	8, 9	mpbir 234	. . 3 ⊢ Tr (TC+ ∪ 𝐴 ∪ 𝐴)
11		ttcmin 36951	. . 3 ⊢ ((𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴) ∧ Tr (TC+ ∪ 𝐴 ∪ 𝐴)) → TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴))
12	1, 10, 11	mp2an 704	. 2 ⊢ TC+ 𝐴 ⊆ (TC+ ∪ 𝐴 ∪ 𝐴)
13		ttcid 36947	. . . . . 6 ⊢ 𝐴 ⊆ TC+ 𝐴
14	13	unissi 4880	. . . . 5 ⊢ ∪ 𝐴 ⊆ ∪ TC+ 𝐴
15		ttctr3 36950	. . . . 5 ⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
16	14, 15	sstri 3945	. . . 4 ⊢ ∪ 𝐴 ⊆ TC+ 𝐴
17		ttcss 36953	. . . 4 ⊢ (∪ 𝐴 ⊆ TC+ 𝐴 → TC+ ∪ 𝐴 ⊆ TC+ 𝐴)
18	16, 17	ax-mp 5	. . 3 ⊢ TC+ ∪ 𝐴 ⊆ TC+ 𝐴
19	18, 13	unssi 4143	. 2 ⊢ (TC+ ∪ 𝐴 ∪ 𝐴) ⊆ TC+ 𝐴
20	12, 19	eqssi 3952	1 ⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)

Syntax hints:   = wceq 1568   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4871  Tr wtr 5217  TC+ cttc 36941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  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 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 36942

This theorem is referenced by:  ttcuni  36968