Theorem chnwrd 18832

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 18831	. . 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 2147  ∀wral 3080   ∖ cdif 4048  {csn 4588   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  (class class class)co 7451  0cc0 11239  1c1 11240   − cmin 11580  Word cword 14699   Chain cchn 18829

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 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574  df-chn 18830

This theorem is used by:  pfxchn  18834  chnexg  18842  chnind  18845  chnub  18846  chnlt  18847  chnccats1  18849  chnccat  18850  chnrev  18851  chnflenfi  18852  chnf  18853  chnpolleha  18856  chnpolfz  18857  fldext2chn  34471  constrextdg2lem  34491  constrext2chnlem  34493  chnsubseqword  48161  chnsubseqwl  48162  chnsubseq  48163  chnsuslle  48164  chnerlem1  48165  chnerlem2  48166  chner  48168  chndin  48174