Theorem oldss 28256

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 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 28220	. . . . . 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 28220	. . . . . 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 3963	. . 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 28223	. . . . . . 7 ⊢ O:On⟶𝒫 No
13	12	fdmi 6721	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2852	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6917	. . . . 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 3952	. . 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 3895  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866  dom cdm 5650   “ cima 5653  Oncon0 6362  ‘cfv 6538  Nocsur 27997  Mcmade 28208  Ocold 28209

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 5326  ax-pr 5390  ax-un 7751

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214

This theorem is used by:  onsbnd  28667