Theorem chnwrd 18696

Description: A chain is an ordered sequence, i.e. a word. (Contributed by Thierry Arnoux, 19-Jun-2025.)

Hypothesis

Ref	Expression
chnwrd.1	⊢ (𝜑 → 𝐶 ∈ ( < Chain 𝐴))

Assertion

Ref	Expression
chnwrd	⊢ (𝜑 → 𝐶 ∈ Word 𝐴)

Proof of Theorem chnwrd

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chnwrd.1	. 2 ⊢ (𝜑 → 𝐶 ∈ ( < Chain 𝐴))
2		ischn 18695	. . 3 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	simplbi 502	. 2 ⊢ (𝐶 ∈ ( < Chain 𝐴) → 𝐶 ∈ Word 𝐴)
4	1, 3	syl 18	1 ⊢ (𝜑 → 𝐶 ∈ Word 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3896  {csn 4584   class class class wbr 5103  dom cdm 5655  ‘cfv 6533  (class class class)co 7413  0cc0 11124  1c1 11125   − cmin 11465  Word cword 14578   Chain cchn 18693

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5665  df-iota 6489  df-fv 6541  df-chn 18694

This theorem is used by:  pfxchn  18698  chnexg  18706  chnind  18709  chnub  18710  chnlt  18711  chnccats1  18713  chnccat  18714  chnrev  18715  chnflenfi  18716  chnf  18717  chnpolleha  18720  chnpolfz  18721  fldext2chn  34238  constrextdg2lem  34258  constrext2chnlem  34260  chnsubseqword  47706  chnsubseqwl  47707  chnsubseq  47708  chnsuslle  47709  chnerlem1  47710  chnerlem2  47711  chner  47713  chndin  47719