Theorem oldss 27827

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 6062	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4877	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	adantl 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
4		oldval 27799	. . . . . 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 27799	. . . . . 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 3999	. . 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 27802	. . . . . . 7 ⊢ O :On⟶𝒫 No
13	12	fdmi 6681	. . . . . 6 ⊢ dom O = On
14	13	eleq2i 2820	. . . . 5 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On)
15		ndmfv 6875	. . . . 5 ⊢ (¬ 𝐴 ∈ dom O → ( O ‘𝐴) = ∅)
16	14, 15	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) = ∅)
17		0ss 4359	. . . 4 ⊢ ∅ ⊆ ( O ‘𝐵)
18	16, 17	eqsstrdi 3988	. . 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 3911  ∅c0 4292  𝒫 cpw 4559  ∪ cuni 4867  dom cdm 5631   “ cima 5634  Oncon0 6320  ‘cfv 6499   No csur 27584   M cmade 27787   O cold 27788

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-1o 8411  df-2o 8412  df-no 27587  df-slt 27588  df-bday 27589  df-sslt 27727  df-scut 27729  df-made 27792  df-old 27793

This theorem is referenced by: (None)