Theorem chnwrd 18543

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 18542	. . 3 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	simplbi 496	. 2 ⊢ (𝐶 ∈ ( < Chain 𝐴) → 𝐶 ∈ Word 𝐴)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Word 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3900  {csn 4582   class class class wbr 5100  dom cdm 5632  ‘cfv 6500  (class class class)co 7368  0cc0 11038  1c1 11039   − cmin 11376  Word cword 14448   Chain cchn 18540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-dm 5642  df-iota 6456  df-fv 6508  df-chn 18541

This theorem is referenced by:  pfxchn  18545  chnexg  18553  chnind  18556  chnub  18557  chnlt  18558  chnccats1  18560  chnccat  18561  chnrev  18562  chnflenfi  18563  chnf  18564  chnpolleha  18567  chnpolfz  18568  fldext2chn  33906  constrextdg2lem  33926  constrext2chnlem  33928  chnsubseqword  47236  chnsubseqwl  47237  chnsubseq  47238  chnsuslle  47239  chnerlem1  47240  chnerlem2  47241  chner  47243