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Theorem rankxplim 9637
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 9640 for the successor case. (Contributed by NM, 19-Sep-2006.)
Hypotheses
Ref Expression
rankxplim.1 𝐴 ∈ V
rankxplim.2 𝐵 ∈ V
Assertion
Ref Expression
rankxplim ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))

Proof of Theorem rankxplim
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwuni 4878 . . . . . . . . . 10 𝑥, 𝑦⟩ ⊆ 𝒫 𝑥, 𝑦
2 vex 3436 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 3436 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3uniop 5429 . . . . . . . . . . 11 𝑥, 𝑦⟩ = {𝑥, 𝑦}
54pweqi 4551 . . . . . . . . . 10 𝒫 𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
61, 5sseqtri 3957 . . . . . . . . 9 𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7 pwuni 4878 . . . . . . . . . . 11 {𝑥, 𝑦} ⊆ 𝒫 {𝑥, 𝑦}
82, 3unipr 4857 . . . . . . . . . . . 12 {𝑥, 𝑦} = (𝑥𝑦)
98pweqi 4551 . . . . . . . . . . 11 𝒫 {𝑥, 𝑦} = 𝒫 (𝑥𝑦)
107, 9sseqtri 3957 . . . . . . . . . 10 {𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦)
1110sspwi 4547 . . . . . . . . 9 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦)
126, 11sstri 3930 . . . . . . . 8 𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦)
132, 3unex 7596 . . . . . . . . . . 11 (𝑥𝑦) ∈ V
1413pwex 5303 . . . . . . . . . 10 𝒫 (𝑥𝑦) ∈ V
1514pwex 5303 . . . . . . . . 9 𝒫 𝒫 (𝑥𝑦) ∈ V
1615rankss 9607 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦) → (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)))
1712, 16ax-mp 5 . . . . . . 7 (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦))
18 rankxplim.1 . . . . . . . . . . 11 𝐴 ∈ V
1918rankel 9597 . . . . . . . . . 10 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
20 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2120rankel 9597 . . . . . . . . . 10 (𝑦𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
222, 3, 18, 20rankelun 9630 . . . . . . . . . 10 (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2319, 21, 22syl2an 596 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2423adantl 482 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
25 ranklim 9602 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
26 ranklim 9602 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2725, 26bitrd 278 . . . . . . . . 9 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2827adantr 481 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2924, 28mpbid 231 . . . . . . 7 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
30 rankon 9553 . . . . . . . 8 (rank‘⟨𝑥, 𝑦⟩) ∈ On
31 rankon 9553 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
32 ontr2 6313 . . . . . . . 8 (((rank‘⟨𝑥, 𝑦⟩) ∈ On ∧ (rank‘(𝐴𝐵)) ∈ On) → (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵))))
3330, 31, 32mp2an 689 . . . . . . 7 (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3417, 29, 33sylancr 587 . . . . . 6 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3530, 31onsucssi 7688 . . . . . 6 ((rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3634, 35sylib 217 . . . . 5 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3736ralrimivva 3123 . . . 4 (Lim (rank‘(𝐴𝐵)) → ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
38 fveq2 6774 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
39 suceq 6331 . . . . . . . 8 ((rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩) → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
4038, 39syl 17 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
4140sseq1d 3952 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵))))
4241ralxp 5750 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
4318, 20xpex 7603 . . . . . 6 (𝐴 × 𝐵) ∈ V
4443rankbnd 9626 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4542, 44bitr3i 276 . . . 4 (∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4637, 45sylib 217 . . 3 (Lim (rank‘(𝐴𝐵)) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4746adantr 481 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4818, 20rankxpl 9633 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
4948adantl 482 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5047, 49eqssd 3938 1 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  Vcvv 3432  cun 3885  wss 3887  c0 4256  𝒫 cpw 4533  {cpr 4563  cop 4567   cuni 4839   × cxp 5587  Oncon0 6266  Lim wlim 6267  suc csuc 6268  cfv 6433  rankcrnk 9521
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523
This theorem is referenced by:  rankxplim3  9639
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