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Theorem rankxplim2 9920
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 6447 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2 n0i 4340 . . . 4 (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
31, 2syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4 df-ne 2941 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5 rankxplim.1 . . . . . . 7 𝐴 ∈ V
6 rankxplim.2 . . . . . . 7 𝐵 ∈ V
75, 6xpex 7773 . . . . . 6 (𝐴 × 𝐵) ∈ V
87rankeq0 9901 . . . . 5 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
98notbii 320 . . . 4 (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
104, 9bitr2i 276 . . 3 (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
113, 10sylib 218 . 2 (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
12 limuni2 6446 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
13 limuni2 6446 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
1412, 13syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
15 rankuni 9903 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
16 rankuni 9903 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1716unieqi 4919 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1815, 17eqtr2i 2766 . . . . 5 (rank‘(𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
19 unixp 6302 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
2019fveq2d 6910 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
2118, 20eqtrid 2789 . . . 4 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
22 limeq 6396 . . . 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, 23imbitrid 244 . 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 206   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cun 3949  c0 4333   cuni 4907   × cxp 5683  Lim wlim 6385  cfv 6561  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  rankxpsuc  9922
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