Theorem chnwrd 18775

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 18774	. . 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 3077   ∖ cdif 3896  {csn 4584   class class class wbr 5103  dom cdm 5651  ‘cfv 6537  (class class class)co 7418  0cc0 11193  1c1 11194   − cmin 11534  Word cword 14651   Chain cchn 18772

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  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 5661  df-iota 6493  df-fv 6545  df-chn 18773

This theorem is used by:  pfxchn  18777  chnexg  18785  chnind  18788  chnub  18789  chnlt  18790  chnccats1  18792  chnccat  18793  chnrev  18794  chnflenfi  18795  chnf  18796  chnpolleha  18799  chnpolfz  18800  fldext2chn  34353  constrextdg2lem  34373  constrext2chnlem  34375  chnsubseqword  47857  chnsubseqwl  47858  chnsubseq  47859  chnsuslle  47860  chnerlem1  47861  chnerlem2  47862  chner  47864  chndin  47870