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Theorem madess 27368
Description: If 𝐴 is less than or equal to ordinal 𝐡, then the made set of 𝐴 is included in the made set of 𝐡. (Contributed by Scott Fenton, 9-Oct-2024.)
Assertion
Ref Expression
madess ((𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
Proof of Theorem madess
Dummy variables π‘Ž 𝑏 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imass2 6101 . . . . . . . . . . 11 (𝐴 βŠ† 𝐡 β†’ ( M β€œ 𝐴) βŠ† ( M β€œ 𝐡))
21unissd 4918 . . . . . . . . . 10 (𝐴 βŠ† 𝐡 β†’ βˆͺ ( M β€œ 𝐴) βŠ† βˆͺ ( M β€œ 𝐡))
32sspwd 4615 . . . . . . . . 9 (𝐴 βŠ† 𝐡 β†’ 𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡))
43adantl 482 . . . . . . . 8 ((𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡) β†’ 𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡))
54adantl 482 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ 𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡))
6 ssrexv 4051 . . . . . . 7 (𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡) β†’ (βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯) β†’ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)))
75, 6syl 17 . . . . . 6 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ (βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯) β†’ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)))
8 ssrexv 4051 . . . . . . . 8 (𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡) β†’ (βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯) β†’ βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)))
95, 8syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ (βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯) β†’ βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)))
109reximdv 3170 . . . . . 6 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ (βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯) β†’ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)))
117, 10syld 47 . . . . 5 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ (βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯) β†’ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)))
1211adantr 481 . . . 4 (((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) ∧ π‘₯ ∈ No ) β†’ (βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯) β†’ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)))
1312ss2rabdv 4073 . . 3 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)} βŠ† {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)})
14 madeval2 27345 . . . 4 (𝐴 ∈ On β†’ ( M β€˜π΄) = {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)})
1514adantr 481 . . 3 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ ( M β€˜π΄) = {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)})
16 madeval2 27345 . . . . 5 (𝐡 ∈ On β†’ ( M β€˜π΅) = {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)})
1716adantr 481 . . . 4 ((𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡) β†’ ( M β€˜π΅) = {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)})
1817adantl 482 . . 3 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ ( M β€˜π΅) = {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐡)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐡)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)})
1913, 15, 183sstr4d 4029 . 2 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
20 madef 27348 . . . . . . 7 M :OnβŸΆπ’« No
2120fdmi 6729 . . . . . 6 dom M = On
2221eleq2i 2825 . . . . 5 (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23 ndmfv 6926 . . . . 5 (Β¬ 𝐴 ∈ dom M β†’ ( M β€˜π΄) = βˆ…)
2422, 23sylnbir 330 . . . 4 (Β¬ 𝐴 ∈ On β†’ ( M β€˜π΄) = βˆ…)
25 0ss 4396 . . . 4 βˆ… βŠ† ( M β€˜π΅)
2624, 25eqsstrdi 4036 . . 3 (Β¬ 𝐴 ∈ On β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
2726adantr 481 . 2 ((Β¬ 𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
2819, 27pm2.61ian 810 1 ((𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148  dom cdm 5676   β€œ cima 5679  Oncon0 6364  β€˜cfv 6543  (class class class)co 7408   No csur 27140   <<s csslt 27279   |s cscut 27281   M cmade 27334
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-1o 8465  df-2o 8466  df-no 27143  df-slt 27144  df-bday 27145  df-sslt 27280  df-scut 27282  df-made 27339
This theorem is referenced by:  oldssmade  27369  madebday  27391
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