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Theorem rankxplim 9876
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 9879 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 4942 . . . . . . . . . 10 ⟨π‘₯, π‘¦βŸ© βŠ† 𝒫 βˆͺ ⟨π‘₯, π‘¦βŸ©
2 vex 3472 . . . . . . . . . . . 12 π‘₯ ∈ V
3 vex 3472 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3uniop 5508 . . . . . . . . . . 11 βˆͺ ⟨π‘₯, π‘¦βŸ© = {π‘₯, 𝑦}
54pweqi 4613 . . . . . . . . . 10 𝒫 βˆͺ ⟨π‘₯, π‘¦βŸ© = 𝒫 {π‘₯, 𝑦}
61, 5sseqtri 4013 . . . . . . . . 9 ⟨π‘₯, π‘¦βŸ© βŠ† 𝒫 {π‘₯, 𝑦}
7 pwuni 4942 . . . . . . . . . . 11 {π‘₯, 𝑦} βŠ† 𝒫 βˆͺ {π‘₯, 𝑦}
82, 3unipr 4919 . . . . . . . . . . . 12 βˆͺ {π‘₯, 𝑦} = (π‘₯ βˆͺ 𝑦)
98pweqi 4613 . . . . . . . . . . 11 𝒫 βˆͺ {π‘₯, 𝑦} = 𝒫 (π‘₯ βˆͺ 𝑦)
107, 9sseqtri 4013 . . . . . . . . . 10 {π‘₯, 𝑦} βŠ† 𝒫 (π‘₯ βˆͺ 𝑦)
1110sspwi 4609 . . . . . . . . 9 𝒫 {π‘₯, 𝑦} βŠ† 𝒫 𝒫 (π‘₯ βˆͺ 𝑦)
126, 11sstri 3986 . . . . . . . 8 ⟨π‘₯, π‘¦βŸ© βŠ† 𝒫 𝒫 (π‘₯ βˆͺ 𝑦)
132, 3unex 7730 . . . . . . . . . . 11 (π‘₯ βˆͺ 𝑦) ∈ V
1413pwex 5371 . . . . . . . . . 10 𝒫 (π‘₯ βˆͺ 𝑦) ∈ V
1514pwex 5371 . . . . . . . . 9 𝒫 𝒫 (π‘₯ βˆͺ 𝑦) ∈ V
1615rankss 9846 . . . . . . . 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 9836 . . . . . . . . . 10 (π‘₯ ∈ 𝐴 β†’ (rankβ€˜π‘₯) ∈ (rankβ€˜π΄))
20 rankxplim.2 . . . . . . . . . . 11 𝐡 ∈ V
2120rankel 9836 . . . . . . . . . 10 (𝑦 ∈ 𝐡 β†’ (rankβ€˜π‘¦) ∈ (rankβ€˜π΅))
222, 3, 18, 20rankelun 9869 . . . . . . . . . 10 (((rankβ€˜π‘₯) ∈ (rankβ€˜π΄) ∧ (rankβ€˜π‘¦) ∈ (rankβ€˜π΅)) β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
2319, 21, 22syl2an 595 . . . . . . . . 9 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡) β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
2423adantl 481 . . . . . . . 8 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)) β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
25 ranklim 9841 . . . . . . . . . 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 9841 . . . . . . . . . 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 279 . . . . . . . . 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 480 . . . . . . . 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 9792 . . . . . . . 8 (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ On
31 rankon 9792 . . . . . . . 8 (rankβ€˜(𝐴 βˆͺ 𝐡)) ∈ On
32 ontr2 6405 . . . . . . . 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 689 . . . . . . 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 586 . . . . . 6 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)) β†’ (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ (rankβ€˜(𝐴 βˆͺ 𝐡)))
3530, 31onsucssi 7827 . . . . . 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 3194 . . . 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 6885 . . . . . . . 8 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ (rankβ€˜π‘§) = (rankβ€˜βŸ¨π‘₯, π‘¦βŸ©))
39 suceq 6424 . . . . . . . 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 4008 . . . . . 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 5835 . . . . 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 7737 . . . . . 6 (𝐴 Γ— 𝐡) ∈ V
4443rankbnd 9865 . . . . 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 277 . . . 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 480 . 2 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (𝐴 Γ— 𝐡) β‰  βˆ…) β†’ (rankβ€˜(𝐴 Γ— 𝐡)) βŠ† (rankβ€˜(𝐴 βˆͺ 𝐡)))
4818, 20rankxpl 9872 . . 3 ((𝐴 Γ— 𝐡) β‰  βˆ… β†’ (rankβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (rankβ€˜(𝐴 Γ— 𝐡)))
4948adantl 481 . 2 ((Lim (rankβ€˜(𝐴 βˆͺ 𝐡)) ∧ (𝐴 Γ— 𝐡) β‰  βˆ…) β†’ (rankβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (rankβ€˜(𝐴 Γ— 𝐡)))
5047, 49eqssd 3994 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  {cpr 4625  βŸ¨cop 4629  βˆͺ cuni 4902   Γ— cxp 5667  Oncon0 6358  Lim wlim 6359  suc csuc 6360  β€˜cfv 6537  rankcrnk 9760
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-r1 9761  df-rank 9762
This theorem is referenced by:  rankxplim3  9878
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