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Theorem rankxplim2 8703
Description: If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)
Hypotheses
Ref Expression
rankxplim.1 𝐴 ∈ V
rankxplim.2 𝐵 ∈ V
Assertion
Ref Expression
rankxplim2 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))

Proof of Theorem rankxplim2
StepHypRef Expression
1 0ellim 5756 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2 n0i 3902 . . . 4 (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
31, 2syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4 df-ne 2791 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5 rankxplim.1 . . . . . . 7 𝐴 ∈ V
6 rankxplim.2 . . . . . . 7 𝐵 ∈ V
75, 6xpex 6927 . . . . . 6 (𝐴 × 𝐵) ∈ V
87rankeq0 8684 . . . . 5 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
98notbii 310 . . . 4 (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
104, 9bitr2i 265 . . 3 (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
113, 10sylib 208 . 2 (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
12 limuni2 5755 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
13 limuni2 5755 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
1412, 13syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
15 rankuni 8686 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
16 rankuni 8686 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1716unieqi 4418 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1815, 17eqtr2i 2644 . . . . 5 (rank‘(𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
19 unixp 5637 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
2019fveq2d 6162 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
2118, 20syl5eq 2667 . . . 4 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
22 limeq 5704 . . . 4 ( (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)) → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴𝐵))))
2321, 22syl 17 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴𝐵))))
2414, 23syl5ib 234 . 2 ((𝐴 × 𝐵) ≠ ∅ → (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵))))
2511, 24mpcom 38 1 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1480  wcel 1987  wne 2790  Vcvv 3190  cun 3558  c0 3897   cuni 4409   × cxp 5082  Lim wlim 5693  cfv 5857  rankcrnk 8586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-reg 8457  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-r1 8587  df-rank 8588
This theorem is referenced by:  rankxpsuc  8705
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