Theorem wess 5617

Description: Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.)

Assertion

Ref	Expression
wess	⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴))

Proof of Theorem wess

Step	Hyp	Ref	Expression
1		frss 5595	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴))
2		soss 5559	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴))
3	1, 2	anim12d 610	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵) → (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)))
4		df-we 5586	. 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵))
5		df-we 5586	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
6	3, 4, 5	3imtr4g 296	1 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3890   Or wor 5538   Fr wfr 5581   We wwe 5583

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3053  df-ss 3907  df-po 5539  df-so 5540  df-fr 5584  df-we 5586

This theorem is referenced by:  wefrc  5625  trssord  6341  ordelord  6346  dford5  7738  omsinds  7838  fnwelem  8081  dfrecs3  8312  ordtypelem8  9440  oismo  9455  cantnfcl  9588  infxpenlem  9935  ac10ct  9956  dfac12lem2  10067  cflim2  10185  cofsmo  10191  hsmexlem1  10348  smobeth  10509  canthwelem  10573  gruina  10741  ltwefz  13925  welb  38057  dnwech  43476  aomclem4  43485  dfac11  43490  oaun3lem1  43802  onfrALTlem3  44971  onfrALTlem3VD  45313