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Theorem rankxplim 9296
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 9299 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 4866 . . . . . . . . . 10 𝑥, 𝑦⟩ ⊆ 𝒫 𝑥, 𝑦
2 vex 3495 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 3495 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3uniop 5396 . . . . . . . . . . 11 𝑥, 𝑦⟩ = {𝑥, 𝑦}
54pweqi 4539 . . . . . . . . . 10 𝒫 𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
61, 5sseqtri 4000 . . . . . . . . 9 𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7 pwuni 4866 . . . . . . . . . . 11 {𝑥, 𝑦} ⊆ 𝒫 {𝑥, 𝑦}
82, 3unipr 4843 . . . . . . . . . . . 12 {𝑥, 𝑦} = (𝑥𝑦)
98pweqi 4539 . . . . . . . . . . 11 𝒫 {𝑥, 𝑦} = 𝒫 (𝑥𝑦)
107, 9sseqtri 4000 . . . . . . . . . 10 {𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦)
11 sspwb 5332 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦) ↔ 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦))
1210, 11mpbi 231 . . . . . . . . 9 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦)
136, 12sstri 3973 . . . . . . . 8 𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦)
142, 3unex 7458 . . . . . . . . . . 11 (𝑥𝑦) ∈ V
1514pwex 5272 . . . . . . . . . 10 𝒫 (𝑥𝑦) ∈ V
1615pwex 5272 . . . . . . . . 9 𝒫 𝒫 (𝑥𝑦) ∈ V
1716rankss 9266 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦) → (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)))
1813, 17ax-mp 5 . . . . . . 7 (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦))
19 rankxplim.1 . . . . . . . . . . 11 𝐴 ∈ V
2019rankel 9256 . . . . . . . . . 10 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
21 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2221rankel 9256 . . . . . . . . . 10 (𝑦𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
232, 3, 19, 21rankelun 9289 . . . . . . . . . 10 (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2420, 22, 23syl2an 595 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2524adantl 482 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
26 ranklim 9261 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
27 ranklim 9261 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2826, 27bitrd 280 . . . . . . . . 9 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2928adantr 481 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
3025, 29mpbid 233 . . . . . . 7 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
31 rankon 9212 . . . . . . . 8 (rank‘⟨𝑥, 𝑦⟩) ∈ On
32 rankon 9212 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
33 ontr2 6231 . . . . . . . 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))))
3431, 32, 33mp2an 688 . . . . . . 7 (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3518, 30, 34sylancr 587 . . . . . 6 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3631, 32onsucssi 7545 . . . . . 6 ((rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3735, 36sylib 219 . . . . 5 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3837ralrimivva 3188 . . . 4 (Lim (rank‘(𝐴𝐵)) → ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
39 fveq2 6663 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
40 suceq 6249 . . . . . . . 8 ((rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩) → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
4139, 40syl 17 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
4241sseq1d 3995 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵))))
4342ralxp 5705 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
4419, 21xpex 7465 . . . . . 6 (𝐴 × 𝐵) ∈ V
4544rankbnd 9285 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4643, 45bitr3i 278 . . . 4 (∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4738, 46sylib 219 . . 3 (Lim (rank‘(𝐴𝐵)) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4847adantr 481 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4919, 21rankxpl 9292 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5049adantl 482 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5148, 50eqssd 3981 1 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492  cun 3931  wss 3933  c0 4288  𝒫 cpw 4535  {cpr 4559  cop 4563   cuni 4830   × cxp 5546  Oncon0 6184  Lim wlim 6185  suc csuc 6186  cfv 6348  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-r1 9181  df-rank 9182
This theorem is referenced by:  rankxplim3  9298
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