Theorem chnrss 18818

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 5152	. . . . 5 ⊢ ( < ⊆ 𝑅 → ((𝑥‘(𝑐 − 1)) < (𝑥‘𝑐) → (𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐)))
2	1	ralimdv 3180	. . . 4 ⊢ ( < ⊆ 𝑅 → (∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1)) < (𝑥‘𝑐) → ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐)))
3	2	anim2d 624	. . 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 18810	. . 3 ⊢ (𝑥 ∈ ( < Chain 𝐴) ↔ (𝑥 ∈ Word 𝐴 ∧ ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1)) < (𝑥‘𝑐)))
5		ischn 18810	. . 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 3919	1 ⊢ ( < ⊆ 𝑅 → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∖ cdif 4048  {csn 4588   class class class wbr 5106  dom cdm 5655  ‘cfv 6558  (class class class)co 7443  0cc0 11227  1c1 11228   − cmin 11568  Word cword 14685   Chain cchn 18808

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-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-dm 5665  df-iota 6512  df-fv 6566  df-chn 18809

This theorem is used by:  chnrdss  18820  chnrrin  48112  chnrun  48114