Theorem madess 27206

Description: If 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, 9-Oct-2024.)

Assertion

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

Proof of Theorem madess

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imass2 6054	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → ( M “ 𝐴) ⊆ ( M “ 𝐵))
2	1	unissd 4875	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ∪ ( M “ 𝐴) ⊆ ∪ ( M “ 𝐵))
3	2	sspwd 4573	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵))
4	3	adantl 482	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → 𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵))
5	4	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → 𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵))
6		ssrexv 4011	. . . . . . 7 ⊢ (𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
8		ssrexv 4011	. . . . . . . 8 ⊢ (𝒫 ∪ ( M “ 𝐴) ⊆ 𝒫 ∪ ( M “ 𝐵) → (∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
9	5, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
10	9	reximdv 3167	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
11	7, 10	syld 47	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
12	11	adantr 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) ∧ 𝑥 ∈ No ) → (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
13	12	ss2rabdv 4033	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)} ⊆ {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
14		madeval2 27183	. . . 4 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
15	14	adantr 481	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
16		madeval2 27183	. . . . 5 ⊢ (𝐵 ∈ On → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
17	16	adantr 481	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
18	17	adantl 482	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐵) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐵)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐵)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
19	13, 15, 18	3sstr4d 3991	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
20		madef 27186	. . . . . . 7 ⊢ M :On⟶𝒫 No
21	20	fdmi 6680	. . . . . 6 ⊢ dom M = On
22	21	eleq2i 2829	. . . . 5 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23		ndmfv 6877	. . . . 5 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
24	22, 23	sylnbir 330	. . . 4 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
25		0ss 4356	. . . 4 ⊢ ∅ ⊆ ( M ‘𝐵)
26	24, 25	eqsstrdi 3998	. . 3 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) ⊆ ( M ‘𝐵))
27	26	adantr 481	. 2 ⊢ ((¬ 𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵)) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
28	19, 27	pm2.61ian 810	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  {crab 3407   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4865   class class class wbr 5105  dom cdm 5633   “ cima 5636  Oncon0 6317  ‘cfv 6496  (class class class)co 7357   No csur 26988   <<s csslt 27120   |s cscut 27122   M cmade 27172

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sslt 27121  df-scut 27123  df-made 27177

This theorem is referenced by:  oldssmade  27207  madebday  27229