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Theorem rankxplim2 8781
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 5825 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2 n0i 3953 . . . 4 (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
31, 2syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4 df-ne 2824 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5 rankxplim.1 . . . . . . 7 𝐴 ∈ V
6 rankxplim.2 . . . . . . 7 𝐵 ∈ V
75, 6xpex 7004 . . . . . 6 (𝐴 × 𝐵) ∈ V
87rankeq0 8762 . . . . 5 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
98notbii 309 . . . 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 5824 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
13 limuni2 5824 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
1412, 13syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
15 rankuni 8764 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
16 rankuni 8764 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1716unieqi 4477 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1815, 17eqtr2i 2674 . . . . 5 (rank‘(𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
19 unixp 5706 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
2019fveq2d 6233 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
2118, 20syl5eq 2697 . . . 4 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
22 limeq 5773 . . . 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 1523  wcel 2030  wne 2823  Vcvv 3231  cun 3605  c0 3948   cuni 4468   × cxp 5141  Lim wlim 5762  cfv 5926  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666
This theorem is referenced by:  rankxpsuc  8783
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