Theorem oldss 28265

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 6057	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (M “ 𝐴) ⊆ (M “ 𝐵))
2	1	unissd 4876	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∪ (M “ 𝐴) ⊆ ∪ (M “ 𝐵))
3	2	adantl 487	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∪ (M “ 𝐴) ⊆ ∪ (M “ 𝐵))
4		oldval 28229	. . . . . 6 ⊢ (𝐴 ∈ On → (O‘𝐴) = ∪ (M “ 𝐴))
5	4	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (O‘𝐴) = ∪ (M “ 𝐴))
6	5	adantr 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (O‘𝐴) = ∪ (M “ 𝐴))
7		oldval 28229	. . . . . 6 ⊢ (𝐵 ∈ On → (O‘𝐵) = ∪ (M “ 𝐵))
8	7	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (O‘𝐵) = ∪ (M “ 𝐵))
9	8	adantr 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (O‘𝐵) = ∪ (M “ 𝐵))
10	3, 6, 9	3sstr4d 3964	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (O‘𝐴) ⊆ (O‘𝐵))
11	10	expl 463	. 2 ⊢ (𝐴 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → (O‘𝐴) ⊆ (O‘𝐵)))
12		oldf 28232	. . . . . . 7 ⊢ O:On⟶𝒫 No
13	12	fdmi 6726	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2852	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6922	. . . . 5 ⊢ (¬ 𝐴 ∈ dom O → (O‘𝐴) = ∅)
16	14, 15	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (O‘𝐴) = ∅)
17		0ss 4349	. . . 4 ⊢ ∅ ⊆ (O‘𝐵)
18	16, 17	eqsstrdi 3953	. . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896  ∅c0 4277  𝒫 cpw 4556  ∪ cuni 4866  dom cdm 5651   “ cima 5654  Oncon0 6365  ‘cfv 6543  Nocsur 28006  Mcmade 28217  Ocold 28218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-1o 8484  df-2o 8485  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-made 28222  df-old 28223

This theorem is used by:  onsbnd  28676