Theorem chnrss 18790

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 5148	. . . . 5 ⊢ ( < ⊆ 𝑅 → ((𝑥‘(𝑐 − 1)) < (𝑥‘𝑐) → (𝑥‘(𝑐 − 1))𝑅(𝑥‘𝑐)))
2	1	ralimdv 3176	. . . 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 18782	. . 3 ⊢ (𝑥 ∈ ( < Chain 𝐴) ↔ (𝑥 ∈ Word 𝐴 ∧ ∀𝑐 ∈ (dom 𝑥 ∖ {0})(𝑥‘(𝑐 − 1)) < (𝑥‘𝑐)))
5		ischn 18782	. . 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 3914	1 ⊢ ( < ⊆ 𝑅 → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∖ cdif 4045  {csn 4583   class class class wbr 5102  dom cdm 5650  ‘cfv 6538  (class class class)co 7420  0cc0 11201  1c1 11202   − cmin 11542  Word cword 14659   Chain cchn 18780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-dm 5660  df-iota 6494  df-fv 6546  df-chn 18781

This theorem is used by:  chnrdss  18792  chnrrin  47983  chnrun  47985