Theorem ttcun 37478

Description: Distribute union of two classes through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcun	⊢ TC+ (𝐴 ∪ 𝐵) = (TC+ 𝐴 ∪ TC+ 𝐵)

Proof of Theorem ttcun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		un4 4176	. 2 ⊢ ((∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ ∪ 𝑥 ∈ 𝐵 TC+ 𝑥) ∪ (𝐴 ∪ 𝐵)) = ((∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ∪ (∪ 𝑥 ∈ 𝐵 TC+ 𝑥 ∪ 𝐵))
2		ttciunun 37477	. . 3 ⊢ TC+ (𝐴 ∪ 𝐵) = (∪ 𝑥 ∈ (𝐴 ∪ 𝐵)TC+ 𝑥 ∪ (𝐴 ∪ 𝐵))
3		iunxun 5057	. . . 4 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)TC+ 𝑥 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ ∪ 𝑥 ∈ 𝐵 TC+ 𝑥)
4	3	uneq1i 4166	. . 3 ⊢ (∪ 𝑥 ∈ (𝐴 ∪ 𝐵)TC+ 𝑥 ∪ (𝐴 ∪ 𝐵)) = ((∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ ∪ 𝑥 ∈ 𝐵 TC+ 𝑥) ∪ (𝐴 ∪ 𝐵))
5	2, 4	eqtri 2786	. 2 ⊢ TC+ (𝐴 ∪ 𝐵) = ((∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ ∪ 𝑥 ∈ 𝐵 TC+ 𝑥) ∪ (𝐴 ∪ 𝐵))
6		ttciunun 37477	. . 3 ⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
7		ttciunun 37477	. . 3 ⊢ TC+ 𝐵 = (∪ 𝑥 ∈ 𝐵 TC+ 𝑥 ∪ 𝐵)
8	6, 7	uneq12i 4168	. 2 ⊢ (TC+ 𝐴 ∪ TC+ 𝐵) = ((∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴) ∪ (∪ 𝑥 ∈ 𝐵 TC+ 𝑥 ∪ 𝐵))
9	1, 5, 8	3eqtr4i 2796	1 ⊢ TC+ (𝐴 ∪ 𝐵) = (TC+ 𝐴 ∪ TC+ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 4152  ∪ ciun 4954  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: (None)