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

Theorem rankxplim 9902
Description: The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 9905 for the successor case. (Contributed by NM, 19-Sep-2006.)
Hypotheses
Ref Expression
rankxplim.1 𝐴 ∈ V
rankxplim.2 𝐡 ∈ V
Assertion
Ref Expression
rankxplim ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (𝐴 Γ— 𝐡) β‰  βˆ…) β†’ (rankβ€˜(𝐴 Γ— 𝐡)) = (rankβ€˜(𝐴 βˆͺ 𝐡)))

Proof of Theorem rankxplim
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwuni 4943 . . . . . . . . . 10 ⟨π‘₯, π‘¦βŸ© βŠ† 𝒫 βˆͺ ⟨π‘₯, π‘¦βŸ©
2 vex 3467 . . . . . . . . . . . 12 π‘₯ ∈ V
3 vex 3467 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3uniop 5511 . . . . . . . . . . 11 βˆͺ ⟨π‘₯, π‘¦βŸ© = {π‘₯, 𝑦}
54pweqi 4614 . . . . . . . . . 10 𝒫 βˆͺ ⟨π‘₯, π‘¦βŸ© = 𝒫 {π‘₯, 𝑦}
61, 5sseqtri 4009 . . . . . . . . 9 ⟨π‘₯, π‘¦βŸ© βŠ† 𝒫 {π‘₯, 𝑦}
7 pwuni 4943 . . . . . . . . . . 11 {π‘₯, 𝑦} βŠ† 𝒫 βˆͺ {π‘₯, 𝑦}
82, 3unipr 4920 . . . . . . . . . . . 12 βˆͺ {π‘₯, 𝑦} = (π‘₯ βˆͺ 𝑦)
98pweqi 4614 . . . . . . . . . . 11 𝒫 βˆͺ {π‘₯, 𝑦} = 𝒫 (π‘₯ βˆͺ 𝑦)
107, 9sseqtri 4009 . . . . . . . . . 10 {π‘₯, 𝑦} βŠ† 𝒫 (π‘₯ βˆͺ 𝑦)
1110sspwi 4610 . . . . . . . . 9 𝒫 {π‘₯, 𝑦} βŠ† 𝒫 𝒫 (π‘₯ βˆͺ 𝑦)
126, 11sstri 3982 . . . . . . . 8 ⟨π‘₯, π‘¦βŸ© βŠ† 𝒫 𝒫 (π‘₯ βˆͺ 𝑦)
132, 3unex 7746 . . . . . . . . . . 11 (π‘₯ βˆͺ 𝑦) ∈ V
1413pwex 5374 . . . . . . . . . 10 𝒫 (π‘₯ βˆͺ 𝑦) ∈ V
1514pwex 5374 . . . . . . . . 9 𝒫 𝒫 (π‘₯ βˆͺ 𝑦) ∈ V
1615rankss 9872 . . . . . . . 8 (⟨π‘₯, π‘¦βŸ© βŠ† 𝒫 𝒫 (π‘₯ βˆͺ 𝑦) β†’ (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)))
1712, 16ax-mp 5 . . . . . . 7 (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦))
18 rankxplim.1 . . . . . . . . . . 11 𝐴 ∈ V
1918rankel 9862 . . . . . . . . . 10 (π‘₯ ∈ 𝐴 β†’ (rankβ€˜π‘₯) ∈ (rankβ€˜π΄))
20 rankxplim.2 . . . . . . . . . . 11 𝐡 ∈ V
2120rankel 9862 . . . . . . . . . 10 (𝑦 ∈ 𝐡 β†’ (rankβ€˜π‘¦) ∈ (rankβ€˜π΅))
222, 3, 18, 20rankelun 9895 . . . . . . . . . 10 (((rankβ€˜π‘₯) ∈ (rankβ€˜π΄) ∧ (rankβ€˜π‘¦) ∈ (rankβ€˜π΅)) β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
2319, 21, 22syl2an 594 . . . . . . . . 9 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡) β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
2423adantl 480 . . . . . . . 8 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)) β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
25 ranklim 9867 . . . . . . . . . 10 (Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) β†’ ((rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ (rankβ€˜π’« (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡))))
26 ranklim 9867 . . . . . . . . . 10 (Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) β†’ ((rankβ€˜π’« (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡))))
2725, 26bitrd 278 . . . . . . . . 9 (Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) β†’ ((rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡))))
2827adantr 479 . . . . . . . 8 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)) β†’ ((rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡))))
2924, 28mpbid 231 . . . . . . 7 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)) β†’ (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
30 rankon 9818 . . . . . . . 8 (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ On
31 rankon 9818 . . . . . . . 8 (rankβ€˜(𝐴 βˆͺ 𝐡)) ∈ On
32 ontr2 6411 . . . . . . . 8 (((rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ On ∧ (rankβ€˜(𝐴 βˆͺ 𝐡)) ∈ On) β†’ (((rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∧ (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡))) β†’ (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡))))
3330, 31, 32mp2an 690 . . . . . . 7 (((rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∧ (rankβ€˜π’« 𝒫 (π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡))) β†’ (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
3417, 29, 33sylancr 585 . . . . . 6 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)) β†’ (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
3530, 31onsucssi 7843 . . . . . 6 ((rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
3634, 35sylib 217 . . . . 5 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)) β†’ suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
3736ralrimivva 3191 . . . 4 (Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
38 fveq2 6892 . . . . . . . 8 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ (rankβ€˜π‘§) = (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©))
39 suceq 6430 . . . . . . . 8 ((rankβ€˜π‘§) = (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) β†’ suc (rankβ€˜π‘§) = suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©))
4038, 39syl 17 . . . . . . 7 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ suc (rankβ€˜π‘§) = suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©))
4140sseq1d 4004 . . . . . 6 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ (suc (rankβ€˜π‘§) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡))))
4241ralxp 5838 . . . . 5 (βˆ€π‘§ ∈ (𝐴 Γ— 𝐡)suc (rankβ€˜π‘§) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
4318, 20xpex 7753 . . . . . 6 (𝐴 Γ— 𝐡) ∈ V
4443rankbnd 9891 . . . . 5 (βˆ€π‘§ ∈ (𝐴 Γ— 𝐡)suc (rankβ€˜π‘§) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ (rankβ€˜(𝐴 Γ— 𝐡)) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
4542, 44bitr3i 276 . . . 4 (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 suc (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)) ↔ (rankβ€˜(𝐴 Γ— 𝐡)) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
4637, 45sylib 217 . . 3 (Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) β†’ (rankβ€˜(𝐴 Γ— 𝐡)) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
4746adantr 479 . 2 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (𝐴 Γ— 𝐡) β‰  βˆ…) β†’ (rankβ€˜(𝐴 Γ— 𝐡)) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
4818, 20rankxpl 9898 . . 3 ((𝐴 Γ— 𝐡) β‰  βˆ… β†’ (rankβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (rankβ€˜(𝐴 Γ— 𝐡)))
4948adantl 480 . 2 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (𝐴 Γ— 𝐡) β‰  βˆ…) β†’ (rankβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (rankβ€˜(𝐴 Γ— 𝐡)))
5047, 49eqssd 3990 1 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (𝐴 Γ— 𝐡) β‰  βˆ…) β†’ (rankβ€˜(𝐴 Γ— 𝐡)) = (rankβ€˜(𝐴 βˆͺ 𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  Vcvv 3463   βˆͺ cun 3937   βŠ† wss 3939  βˆ…c0 4318  π’« cpw 4598  {cpr 4626  βŸ¨cop 4630  βˆͺ cuni 4903   Γ— cxp 5670  Oncon0 6364  Lim wlim 6365  suc csuc 6366  β€˜cfv 6543  rankcrnk 9786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-reg 9615  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  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-ov 7419  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-r1 9787  df-rank 9788
This theorem is referenced by:  rankxplim3  9904
  Copyright terms: Public domain W3C validator