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Theorem rankxplim 8625
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 8628 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 4825 . . . . . . . . . 10 𝑥, 𝑦⟩ ⊆ 𝒫 𝑥, 𝑦
2 vex 3176 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 3176 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3uniop 4902 . . . . . . . . . . 11 𝑥, 𝑦⟩ = {𝑥, 𝑦}
54pweqi 4112 . . . . . . . . . 10 𝒫 𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
61, 5sseqtri 3600 . . . . . . . . 9 𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7 pwuni 4825 . . . . . . . . . . 11 {𝑥, 𝑦} ⊆ 𝒫 {𝑥, 𝑦}
82, 3unipr 4385 . . . . . . . . . . . 12 {𝑥, 𝑦} = (𝑥𝑦)
98pweqi 4112 . . . . . . . . . . 11 𝒫 {𝑥, 𝑦} = 𝒫 (𝑥𝑦)
107, 9sseqtri 3600 . . . . . . . . . 10 {𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦)
11 sspwb 4844 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝒫 (𝑥𝑦) ↔ 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦))
1210, 11mpbi 219 . . . . . . . . 9 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥𝑦)
136, 12sstri 3577 . . . . . . . 8 𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥𝑦)
142, 3unex 6854 . . . . . . . . . . 11 (𝑥𝑦) ∈ V
1514pwex 4774 . . . . . . . . . 10 𝒫 (𝑥𝑦) ∈ V
1615pwex 4774 . . . . . . . . 9 𝒫 𝒫 (𝑥𝑦) ∈ V
1716rankss 8595 . . . . . . . 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 8585 . . . . . . . . . 10 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
21 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2221rankel 8585 . . . . . . . . . 10 (𝑦𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
232, 3, 19, 21rankelun 8618 . . . . . . . . . 10 (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2420, 22, 23syl2an 493 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
2524adantl 481 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
26 ranklim 8590 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
27 ranklim 8590 . . . . . . . . . 10 (Lim (rank‘(𝐴𝐵)) → ((rank‘𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2826, 27bitrd 267 . . . . . . . . 9 (Lim (rank‘(𝐴𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
2928adantr 480 . . . . . . . 8 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → ((rank‘(𝑥𝑦)) ∈ (rank‘(𝐴𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))))
3025, 29mpbid 221 . . . . . . 7 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵)))
31 rankon 8541 . . . . . . . 8 (rank‘⟨𝑥, 𝑦⟩) ∈ On
32 rankon 8541 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
33 ontr2 5689 . . . . . . . 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 704 . . . . . . 7 (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥𝑦)) ∈ (rank‘(𝐴𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3518, 30, 34sylancr 694 . . . . . 6 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)))
3631, 32onsucssi 6933 . . . . . 6 ((rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3735, 36sylib 207 . . . . 5 ((Lim (rank‘(𝐴𝐵)) ∧ (𝑥𝐴𝑦𝐵)) → suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
3837ralrimivva 2954 . . . 4 (Lim (rank‘(𝐴𝐵)) → ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
39 fveq2 6103 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
40 suceq 5707 . . . . . . . 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 3595 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵))))
4342ralxp 5185 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)))
4419, 21xpex 6860 . . . . . 6 (𝐴 × 𝐵) ∈ V
4544rankbnd 8614 . . . . 5 (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4643, 45bitr3i 265 . . . 4 (∀𝑥𝐴𝑦𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4738, 46sylib 207 . . 3 (Lim (rank‘(𝐴𝐵)) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4847adantr 480 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴𝐵)))
4919, 21rankxpl 8621 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5049adantl 481 . 2 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
5148, 50eqssd 3585 1 ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  Vcvv 3173  cun 3538  wss 3540  c0 3874  𝒫 cpw 4108  {cpr 4127  cop 4131   cuni 4372   × cxp 5036  Oncon0 5640  Lim wlim 5641  suc csuc 5642  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:  rankxplim3  8627
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