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Theorem rankxplim 9880
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 9883 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 4949 . . . . . . . . . 10 𝑥, 𝑦⟩ ⊆ 𝒫 𝑥, 𝑦
2 vex 3477 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 3477 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3uniop 5515 . . . . . . . . . . 11 𝑥, 𝑦⟩ = {𝑥, 𝑦}
54pweqi 4618 . . . . . . . . . 10 𝒫 𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
61, 5sseqtri 4018 . . . . . . . . 9 𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7 pwuni 4949 . . . . . . . . . . 11 {𝑥, 𝑦} ⊆ 𝒫 {𝑥, 𝑦}
82, 3unipr 4926 . . . . . . . . . . . 12 {𝑥, 𝑦} = (𝑥𝑦)
98pweqi 4618 . . . . . . . . . . 11 𝒫 {𝑥, 𝑦} = 𝒫 (𝑥𝑦)
107, 9sseqtri 4018 . . . . . . . . . 10 {𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦)
1110sspwi 4614 . . . . . . . . 9 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦)
126, 11sstri 3991 . . . . . . . 8 𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦)
132, 3unex 7737 . . . . . . . . . . 11 (𝑥𝑦) ∈ V
1413pwex 5378 . . . . . . . . . 10 𝒫 (𝑥𝑦) ∈ V
1514pwex 5378 . . . . . . . . 9 𝒫 𝒫 (𝑥𝑦) ∈ V
1615rankss 9850 . . . . . . . 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 9840 . . . . . . . . . 10 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
20 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2120rankel 9840 . . . . . . . . . 10 (𝑦𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
222, 3, 18, 20rankelun 9873 . . . . . . . . . 10 (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2319, 21, 22syl2an 595 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2423adantl 481 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
25 ranklim 9845 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
26 ranklim 9845 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2725, 26bitrd 279 . . . . . . . . 9 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2827adantr 480 . . . . . . . 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 9796 . . . . . . . 8 (rank‘⟨𝑥, 𝑦⟩) ∈ On
31 rankon 9796 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
32 ontr2 6411 . . . . . . . 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 586 . . . . . 6 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3530, 31onsucssi 7834 . . . . . 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 3199 . . . 4 (Lim (rank‘(𝐴𝐵)) → ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
38 fveq2 6891 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
39 suceq 6430 . . . . . . . 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 4013 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵))))
4241ralxp 5841 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
4318, 20xpex 7744 . . . . . 6 (𝐴 × 𝐵) ∈ V
4443rankbnd 9869 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4542, 44bitr3i 277 . . . 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 480 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4818, 20rankxpl 9876 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
4948adantl 481 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5047, 49eqssd 3999 1 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  Vcvv 3473  cun 3946  wss 3948  c0 4322  𝒫 cpw 4602  {cpr 4630  cop 4634   cuni 4908   × cxp 5674  Oncon0 6364  Lim wlim 6365  suc csuc 6366  cfv 6543  rankcrnk 9764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-reg 9593  ax-inf2 9642
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-r1 9765  df-rank 9766
This theorem is referenced by:  rankxplim3  9882
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