Theorem oldss 27880

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 6054	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4848	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	adantl 482	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
4		oldval 27844	. . . . . 6 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
5	4	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
6	5	adantr 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
7		oldval 27844	. . . . . 6 ⊢ (𝐵 ∈ On → ( O ‘𝐵) = ∪ ( M “ 𝐵))
8	7	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( O ‘𝐵) = ∪ ( M “ 𝐵))
9	8	adantr 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐵) = ∪ ( M “ 𝐵))
10	3, 6, 9	3sstr4d 3970	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))
11	10	expl 458	. 2 ⊢ (𝐴 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵)))
12		oldf 27847	. . . . . . 7 ⊢ O :On⟶𝒫 No
13	12	fdmi 6666	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2831	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6859	. . . . 5 ⊢ (¬ 𝐴 ∈ dom O → ( O ‘𝐴) = ∅)
16	14, 15	sylnbir 332	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) = ∅)
17		0ss 4328	. . . 4 ⊢ ∅ ⊆ ( O ‘𝐵)
18	16, 17	eqsstrdi 3959	. . 3 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) ⊆ ( O ‘𝐵))
19	18	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵)))
20	11, 19	pm2.61i 183	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ∪ cuni 4838  dom cdm 5618   “ cima 5621  Oncon0 6310  ‘cfv 6485   No csur 27621   M cmade 27832   O cold 27833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-no 27624  df-lts 27625  df-bday 27626  df-slts 27768  df-cuts 27770  df-made 27837  df-old 27838

This theorem is referenced by:  onsbnd  28291