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Theorem rankxplim 9041
Description: The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 9044 for the successor case. (Contributed by NM, 19-Sep-2006.)
Hypotheses
Ref Expression
rankxplim.1 𝐴 ∈ V
rankxplim.2 𝐵 ∈ V
Assertion
Ref Expression
rankxplim ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))

Proof of Theorem rankxplim
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwuni 4711 . . . . . . . . . 10 𝑥, 𝑦⟩ ⊆ 𝒫 𝑥, 𝑦
2 vex 3401 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 3401 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3uniop 5214 . . . . . . . . . . 11 𝑥, 𝑦⟩ = {𝑥, 𝑦}
54pweqi 4383 . . . . . . . . . 10 𝒫 𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
61, 5sseqtri 3856 . . . . . . . . 9 𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7 pwuni 4711 . . . . . . . . . . 11 {𝑥, 𝑦} ⊆ 𝒫 {𝑥, 𝑦}
82, 3unipr 4686 . . . . . . . . . . . 12 {𝑥, 𝑦} = (𝑥𝑦)
98pweqi 4383 . . . . . . . . . . 11 𝒫 {𝑥, 𝑦} = 𝒫 (𝑥𝑦)
107, 9sseqtri 3856 . . . . . . . . . 10 {𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦)
11 sspwb 5151 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦) ↔ 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦))
1210, 11mpbi 222 . . . . . . . . 9 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦)
136, 12sstri 3830 . . . . . . . 8 𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦)
142, 3unex 7235 . . . . . . . . . . 11 (𝑥𝑦) ∈ V
1514pwex 5094 . . . . . . . . . 10 𝒫 (𝑥𝑦) ∈ V
1615pwex 5094 . . . . . . . . 9 𝒫 𝒫 (𝑥𝑦) ∈ V
1716rankss 9011 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦) → (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)))
1813, 17ax-mp 5 . . . . . . 7 (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦))
19 rankxplim.1 . . . . . . . . . . 11 𝐴 ∈ V
2019rankel 9001 . . . . . . . . . 10 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
21 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2221rankel 9001 . . . . . . . . . 10 (𝑦𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
232, 3, 19, 21rankelun 9034 . . . . . . . . . 10 (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2420, 22, 23syl2an 589 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2524adantl 475 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
26 ranklim 9006 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
27 ranklim 9006 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2826, 27bitrd 271 . . . . . . . . 9 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2928adantr 474 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
3025, 29mpbid 224 . . . . . . 7 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
31 rankon 8957 . . . . . . . 8 (rank‘⟨𝑥, 𝑦⟩) ∈ On
32 rankon 8957 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
33 ontr2 6025 . . . . . . . 8 (((rank‘⟨𝑥, 𝑦⟩) ∈ On ∧ (rank‘(𝐴𝐵)) ∈ On) → (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵))))
3431, 32, 33mp2an 682 . . . . . . 7 (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3518, 30, 34sylancr 581 . . . . . 6 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3631, 32onsucssi 7321 . . . . . 6 ((rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3735, 36sylib 210 . . . . 5 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3837ralrimivva 3153 . . . 4 (Lim (rank‘(𝐴𝐵)) → ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
39 fveq2 6448 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
40 suceq 6043 . . . . . . . 8 ((rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩) → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
4139, 40syl 17 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
4241sseq1d 3851 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵))))
4342ralxp 5511 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
4419, 21xpex 7242 . . . . . 6 (𝐴 × 𝐵) ∈ V
4544rankbnd 9030 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4643, 45bitr3i 269 . . . 4 (∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4738, 46sylib 210 . . 3 (Lim (rank‘(𝐴𝐵)) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4847adantr 474 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4919, 21rankxpl 9037 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5049adantl 475 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5148, 50eqssd 3838 1 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wne 2969  wral 3090  Vcvv 3398  cun 3790  wss 3792  c0 4141  𝒫 cpw 4379  {cpr 4400  cop 4404   cuni 4673   × cxp 5355  Oncon0 5978  Lim wlim 5979  suc csuc 5980  cfv 6137  rankcrnk 8925
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 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-reg 8788  ax-inf2 8837
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 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-r1 8926  df-rank 8927
This theorem is referenced by:  rankxplim3  9043
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