Theorem chnrss 18661

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 5149	. . . . 5 ⊢ ( < ⊆ 𝑅 → ((𝑥‘(𝑐 − 1)) < (𝑥‘𝑐) → (𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐)))
2	1	ralimdv 3179	. . . 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 18653	. . 3 ⊢ (𝑥 ∈ ( < Chain 𝐴) ↔ (𝑥 ∈ Word 𝐴 ∧ ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1)) < (𝑥‘𝑐)))
5		ischn 18653	. . 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 3945	1 ⊢ ( < ⊆ 𝑅 → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  ∀wral 3079   ∖ cdif 3904   ⊆ wss 3907  {csn 4585   class class class wbr 5105  dom cdm 5652  ‘cfv 6525  (class class class)co 7400  0cc0 11088  1c1 11089   − cmin 11429  Word cword 14540   Chain cchn 18651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-dm 5662  df-iota 6481  df-fv 6533  df-chn 18652

This theorem is referenced by:  chnrdss  18663