Theorem rightssold 33653

Description: The right options are a subset of the old set. (Contributed by Scott Fenton, 7-Aug-2024.)

Assertion

Ref	Expression
rightssold	⊢ (𝑋 ∈ No → ( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋)))

Proof of Theorem rightssold

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rightval 33638	. 2 ⊢ (𝑋 ∈ No → ( R ‘𝑋) = {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑥})
2		ssrab2 3986	. 2 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑥} ⊆ ( O ‘( bday ‘𝑋))
3	1, 2	eqsstrdi 3948	1 ⊢ (𝑋 ∈ No → ( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋)))

Syntax hints:   → wi 4   ∈ wcel 2111  {crab 3074   ⊆ wss 3860   class class class wbr 5036  ‘cfv 6340   No csur 33440   <s cslt 33441   bday cbday 33442   O cold 33621   R cright 33624

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-right 33629

This theorem is referenced by:  rightssno  33655  rightirr  33665  right0s  33667  madebday  33671  lrrecpred  33683