Theorem oldss 27823

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 6050	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4866	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	adantl 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
4		oldval 27795	. . . . . 6 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
5	4	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
6	5	adantr 480	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
7		oldval 27795	. . . . . 6 ⊢ (𝐵 ∈ On → ( O ‘𝐵) = ∪ ( M “ 𝐵))
8	7	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ( O ‘𝐵) = ∪ ( M “ 𝐵))
9	8	adantr 480	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐵) = ∪ ( M “ 𝐵))
10	3, 6, 9	3sstr4d 3985	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))
11	10	expl 457	. 2 ⊢ (𝐴 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵)))
12		oldf 27798	. . . . . . 7 ⊢ O :On⟶𝒫 No
13	12	fdmi 6662	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2823	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6854	. . . . 5 ⊢ (¬ 𝐴 ∈ dom O → ( O ‘𝐴) = ∅)
16	14, 15	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) = ∅)
17		0ss 4347	. . . 4 ⊢ ∅ ⊆ ( O ‘𝐵)
18	16, 17	eqsstrdi 3974	. . 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 182	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  ∪ cuni 4856  dom cdm 5614   “ cima 5617  Oncon0 6306  ‘cfv 6481   No csur 27578   M cmade 27783   O cold 27784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-no 27581  df-slt 27582  df-bday 27583  df-sslt 27721  df-scut 27723  df-made 27788  df-old 27789

This theorem is referenced by: (None)