Theorem oldss 28313

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 6073	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (M “ 𝐴) ⊆ (M “ 𝐵))
2	1	unissd 4882	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∪ (M “ 𝐴) ⊆ ∪ (M “ 𝐵))
3	2	adantl 487	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∪ (M “ 𝐴) ⊆ ∪ (M “ 𝐵))
4		oldval 28277	. . . . . 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 28277	. . . . . 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 3968	. . 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 28280	. . . . . . 7 ⊢ O:On⟶𝒫 No
13	12	fdmi 6751	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2856	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6948	. . . . 5 ⊢ (¬ 𝐴 ∈ dom O → (O‘𝐴) = ∅)
16	14, 15	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (O‘𝐴) = ∅)
17		0ss 4353	. . . 4 ⊢ ∅ ⊆ (O‘𝐵)
18	16, 17	eqsstrdi 3957	. . 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 1570   ∈ wcel 2147   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4872  dom cdm 5658   “ cima 5661  Oncon0 6388  ‘cfv 6566  Nocsur 28054  Mcmade 28265  Ocold 28266

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271

This theorem is used by:  onsbnd  28724