Theorem madess 33650

Description: If ordinal 𝐴 is less than or equal to ordinal 𝐵, then the made set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 7-Aug-2024.)

Assertion

Ref	Expression
madess	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))

Proof of Theorem madess

Dummy variables 𝑥 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imass2 5942	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	3ad2ant3 1132	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M “ 𝐴) ⊆ ( M “ 𝐵))
3	2	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ No ) → ( M “ 𝐴) ⊆ ( M “ 𝐵))
4	3	unissd 4811	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ No ) → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
5	4	sspwd 4512	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ No ) → 𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵))
6		ssrexv 3961	. . . . . 6 ⊢ (𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵) → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
7	5, 6	syl 17	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ No ) → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
8		ssrexv 3961	. . . . . . 7 ⊢ (𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵) → (∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
9	5, 8	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ No ) → (∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
10	9	reximdv 3197	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ No ) → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
11	7, 10	syld 47	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ No ) → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
12	11	ralrimiva 3113	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ∀𝑥 ∈ No (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
13		ss2rab 3977	. . 3 ⊢ ({𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} ⊆ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} ↔ ∀𝑥 ∈ No (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
14	12, 13	sylibr 237	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} ⊆ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
15		madeval2 33631	. . 3 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
16	15	3ad2ant1 1130	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
17		madeval2 33631	. . 3 ⊢ (𝐵 ∈ On → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
18	17	3ad2ant2 1131	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
19	14, 16, 18	3sstr4d 3941	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074   ⊆ wss 3860  𝒫 cpw 4497  ∪ cuni 4801   class class class wbr 5036   “ cima 5531  Oncon0 6174  ‘cfv 6340  (class class class)co 7156   No csur 33440   <<s csslt 33572   |s cscut 33574   M cmade 33620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-wrecs 7963  df-recs 8024  df-1o 8118  df-2o 8119  df-no 33443  df-slt 33444  df-bday 33445  df-sslt 33573  df-scut 33575  df-made 33625

This theorem is referenced by:  oldssmade  33651  madebday  33671