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Theorem madess 27372
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 6102 . . . . . . . . . . 11 (𝐴 βŠ† 𝐡 β†’ ( M β€œ 𝐴) βŠ† ( M β€œ 𝐡))
21unissd 4919 . . . . . . . . . 10 (𝐴 βŠ† 𝐡 β†’ βˆͺ ( M β€œ 𝐴) βŠ† βˆͺ ( M β€œ 𝐡))
32sspwd 4616 . . . . . . . . 9 (𝐴 βŠ† 𝐡 β†’ 𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡))
43adantl 483 . . . . . . . 8 ((𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡) β†’ 𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡))
54adantl 483 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ 𝒫 βˆͺ ( M β€œ 𝐴) βŠ† 𝒫 βˆͺ ( M β€œ 𝐡))
6 ssrexv 4052 . . . . . . 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 4052 . . . . . . . 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 3171 . . . . . 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 482 . . . 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 4074 . . 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 27349 . . . 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 482 . . 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 27349 . . . . 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 482 . . . 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 483 . . 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 4030 . 2 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
20 madef 27352 . . . . . . 7 M :OnβŸΆπ’« No
2120fdmi 6730 . . . . . 6 dom M = On
2221eleq2i 2826 . . . . 5 (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23 ndmfv 6927 . . . . 5 (Β¬ 𝐴 ∈ dom M β†’ ( M β€˜π΄) = βˆ…)
2422, 23sylnbir 331 . . . 4 (Β¬ 𝐴 ∈ On β†’ ( M β€˜π΄) = βˆ…)
25 0ss 4397 . . . 4 βˆ… βŠ† ( M β€˜π΅)
2624, 25eqsstrdi 4037 . . 3 (Β¬ 𝐴 ∈ On β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
2726adantr 482 . 2 ((Β¬ 𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
2819, 27pm2.61ian 811 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149  dom cdm 5677   β€œ cima 5680  Oncon0 6365  β€˜cfv 6544  (class class class)co 7409   No csur 27143   <<s csslt 27282   |s cscut 27284   M cmade 27338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147  df-bday 27148  df-sslt 27283  df-scut 27285  df-made 27343
This theorem is referenced by:  oldssmade  27373  madebday  27395
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