Theorem oldss 28028

Description: If 𝐴 is less than or equal to ordinal 𝐵, then the old set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.)

Assertion

Ref	Expression
oldss	⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))

Proof of Theorem oldss

Step	Hyp	Ref	Expression
1		imass2 6105	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4886	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	adantl 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
4		oldval 27992	. . . . . 6 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
5	4	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
6	5	adantr 485	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
7		oldval 27992	. . . . . 6 ⊢ (𝐵 ∈ On → ( O ‘𝐵) = ∪ ( M “ 𝐵))
8	7	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( O ‘𝐵) = ∪ ( M “ 𝐵))
9	8	adantr 485	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐵) = ∪ ( M “ 𝐵))
10	3, 6, 9	3sstr4d 4000	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))
11	10	expl 462	. 2 ⊢ (𝐴 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵)))
12		oldf 27995	. . . . . . 7 ⊢ O :On⟶𝒫 No
13	12	fdmi 6718	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2861	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6914	. . . . 5 ⊢ (¬ 𝐴 ∈ dom O → ( O ‘𝐴) = ∅)
16	14, 15	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) = ∅)
17		0ss 4364	. . . 4 ⊢ ∅ ⊆ ( O ‘𝐵)
18	16, 17	eqsstrdi 3989	. . 3 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) ⊆ ( O ‘𝐵))
19	18	a1d 26	. 2 ⊢ (¬ 𝐴 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵)))
20	11, 19	pm2.61i 184	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876  dom cdm 5662   “ cima 5665  Oncon0 6361  ‘cfv 6537   No csur 27769   M cmade 27980   O cold 27981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-1o 8452  df-2o 8453  df-no 27772  df-lts 27773  df-bday 27774  df-slts 27916  df-cuts 27918  df-made 27985  df-old 27986

This theorem is referenced by:  onsbnd  28439