Theorem wess 5608

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 5586	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴))
2		soss 5550	. . 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 5577	. 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵))
5		df-we 5577	. 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 5529   Fr wfr 5572   We wwe 5574

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 5530  df-so 5531  df-fr 5575  df-we 5577

This theorem is referenced by:  wefrc  5616  trssord  6332  ordelord  6337  dford5  7729  omsinds  7829  fnwelem  8072  dfrecs3  8303  ordtypelem8  9431  oismo  9446  cantnfcl  9577  infxpenlem  9924  ac10ct  9945  dfac12lem2  10056  cflim2  10174  cofsmo  10180  hsmexlem1  10337  smobeth  10498  canthwelem  10562  gruina  10730  ltwefz  13914  welb  38068  dnwech  43491  aomclem4  43500  dfac11  43505  oaun3lem1  43817  onfrALTlem3  44986  onfrALTlem3VD  45328