Theorem chnrss 18677

Description: Chains under a relation are also chains under any superset relation. (Contributed by Ender Ting, 20-Jan-2026.)

Assertion

Ref	Expression
chnrss	⊢ ( < ⊆ 𝑅 → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐴))

Proof of Theorem chnrss

Dummy variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssbr 5154	. . . . 5 ⊢ ( < ⊆ 𝑅 → ((𝑥‘(𝑐 − 1)) < (𝑥‘𝑐) → (𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐)))
2	1	ralimdv 3178	. . . 4 ⊢ ( < ⊆ 𝑅 → (∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1)) < (𝑥‘𝑐) → ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐)))
3	2	anim2d 623	. . 3 ⊢ ( < ⊆ 𝑅 → ((𝑥 ∈ Word 𝐴 ∧ ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1)) < (𝑥‘𝑐)) → (𝑥 ∈ Word 𝐴 ∧ ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐))))
4		ischn 18669	. . 3 ⊢ (𝑥 ∈ ( < Chain 𝐴) ↔ (𝑥 ∈ Word 𝐴 ∧ ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1)) < (𝑥‘𝑐)))
5		ischn 18669	. . 3 ⊢ (𝑥 ∈ (𝑅 Chain 𝐴) ↔ (𝑥 ∈ Word 𝐴 ∧ ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐)))
6	3, 4, 5	3imtr4g 299	. 2 ⊢ ( < ⊆ 𝑅 → (𝑥 ∈ ( < Chain 𝐴) → 𝑥 ∈ (𝑅 Chain 𝐴)))
7	6	ssrdv 3942	1 ⊢ ( < ⊆ 𝑅 → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  ∀wral 3078   ∖ cdif 3901   ⊆ wss 3904  {csn 4588   class class class wbr 5108  dom cdm 5660  ‘cfv 6536  (class class class)co 7412  0cc0 11106  1c1 11107   − cmin 11447  Word cword 14557   Chain cchn 18667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-dm 5670  df-iota 6492  df-fv 6544  df-chn 18668

This theorem is used by:  chnrdss  18679