MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  madess Structured version   Visualization version   GIF version

Theorem madess 27235
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 6058 . . . . . . . . . . 11 (𝐴 βŠ† 𝐡 β†’ ( M β€œ 𝐴) βŠ† ( M β€œ 𝐡))
21unissd 4879 . . . . . . . . . 10 (𝐴 βŠ† 𝐡 β†’ βˆͺ ( M β€œ 𝐴) βŠ† βˆͺ ( M β€œ 𝐡))
32sspwd 4577 . . . . . . . . 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 4015 . . . . . . 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 4015 . . . . . . . 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 3164 . . . . . 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 4037 . . 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 27212 . . . 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 27212 . . . . 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 3995 . 2 ((𝐴 ∈ On ∧ (𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡)) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
20 madef 27215 . . . . . . 7 M :OnβŸΆπ’« No
2120fdmi 6684 . . . . . 6 dom M = On
2221eleq2i 2826 . . . . 5 (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
23 ndmfv 6881 . . . . 5 (Β¬ 𝐴 ∈ dom M β†’ ( M β€˜π΄) = βˆ…)
2422, 23sylnbir 331 . . . 4 (Β¬ 𝐴 ∈ On β†’ ( M β€˜π΄) = βˆ…)
25 0ss 4360 . . . 4 βˆ… βŠ† ( M β€˜π΅)
2624, 25eqsstrdi 4002 . . 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 3070  {crab 3406   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  βˆͺ cuni 4869   class class class wbr 5109  dom cdm 5637   β€œ cima 5640  Oncon0 6321  β€˜cfv 6500  (class class class)co 7361   No csur 27011   <<s csslt 27149   |s cscut 27151   M cmade 27201
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-1o 8416  df-2o 8417  df-no 27014  df-slt 27015  df-bday 27016  df-sslt 27150  df-scut 27152  df-made 27206
This theorem is referenced by:  oldssmade  27236  madebday  27258
  Copyright terms: Public domain W3C validator