MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankxplim2 Structured version   Visualization version   GIF version

Theorem rankxplim2 9040
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 6038 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2 n0i 4148 . . . 4 (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
31, 2syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4 df-ne 2970 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5 rankxplim.1 . . . . . . 7 𝐴 ∈ V
6 rankxplim.2 . . . . . . 7 𝐵 ∈ V
75, 6xpex 7240 . . . . . 6 (𝐴 × 𝐵) ∈ V
87rankeq0 9021 . . . . 5 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
98notbii 312 . . . 4 (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
104, 9bitr2i 268 . . 3 (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
113, 10sylib 210 . 2 (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
12 limuni2 6037 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
13 limuni2 6037 . . . 4 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
1412, 13syl 17 . . 3 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
15 rankuni 9023 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
16 rankuni 9023 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1716unieqi 4680 . . . . . 6 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
1815, 17eqtr2i 2803 . . . . 5 (rank‘(𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
19 unixp 5922 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
2019fveq2d 6450 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
2118, 20syl5eq 2826 . . . 4 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
22 limeq 5988 . . . 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 236 . 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 198   = wceq 1601  wcel 2107  wne 2969  Vcvv 3398  cun 3790  c0 4141   cuni 4671   × cxp 5353  Lim wlim 5977  cfv 6135  rankcrnk 8923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-reg 8786  ax-inf2 8835
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-r1 8924  df-rank 8925
This theorem is referenced by:  rankxpsuc  9042
  Copyright terms: Public domain W3C validator