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Theorem rankxplim2 8626
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 5704 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2 n0i 3879 . . . 4 (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
31, 2syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4 df-ne 2782 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5 rankxplim.1 . . . . . . 7 𝐴 ∈ V
6 rankxplim.2 . . . . . . 7 𝐵 ∈ V
75, 6xpex 6860 . . . . . 6 (𝐴 × 𝐵) ∈ V
87rankeq0 8607 . . . . 5 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
98notbii 309 . . . 4 (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
104, 9bitr2i 264 . . 3 (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
113, 10sylib 207 . 2 (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
12 limuni2 5703 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
13 limuni2 5703 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
1412, 13syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
15 rankuni 8609 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
16 rankuni 8609 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1716unieqi 4381 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1815, 17eqtr2i 2633 . . . . 5 (rank‘(𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
19 unixp 5585 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
2019fveq2d 6107 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
2118, 20syl5eq 2656 . . . 4 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
22 limeq 5652 . . . 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 233 . 2 ((𝐴 × 𝐵) ≠ ∅ → (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵))))
2511, 24mpcom 37 1 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  cun 3538  c0 3874   cuni 4372   × cxp 5036  Lim wlim 5641  cfv 5804  rankcrnk 8509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511
This theorem is referenced by:  rankxpsuc  8628
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