Theorem oldss 27792

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 6053	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4868	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	adantl 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
4		oldval 27764	. . . . . 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 27764	. . . . . 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 3991	. . 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 27767	. . . . . . 7 ⊢ O :On⟶𝒫 No
13	12	fdmi 6663	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2820	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6855	. . . . 5 ⊢ (¬ 𝐴 ∈ dom O → ( O ‘𝐴) = ∅)
16	14, 15	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) = ∅)
17		0ss 4351	. . . 4 ⊢ ∅ ⊆ ( O ‘𝐵)
18	16, 17	eqsstrdi 3980	. . 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 1540   ∈ wcel 2109   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858  dom cdm 5619   “ cima 5622  Oncon0 6307  ‘cfv 6482   No csur 27549   M cmade 27752   O cold 27753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-1o 8388  df-2o 8389  df-no 27552  df-slt 27553  df-bday 27554  df-sslt 27692  df-scut 27694  df-made 27757  df-old 27758

This theorem is referenced by: (None)