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Theorem rankxpsuc 9922
Description: The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 9919 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)
Hypotheses
Ref Expression
rankxplim.1 𝐴 ∈ V
rankxplim.2 𝐵 ∈ V
Assertion
Ref Expression
rankxpsuc (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴𝐵)))

Proof of Theorem rankxpsuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 unixp 6302 . . . . . . . 8 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
21fveq2d 6910 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
3 rankuni 9903 . . . . . . . 8 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
4 rankuni 9903 . . . . . . . . 9 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
54unieqi 4919 . . . . . . . 8 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
63, 5eqtri 2765 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
72, 6eqtr3di 2792 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
8 suc11reg 9659 . . . . . 6 (suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)) ↔ (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
97, 8sylibr 234 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)))
109adantl 481 . . . 4 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)))
11 fvex 6919 . . . . . . . . . . . . . 14 (rank‘(𝐴𝐵)) ∈ V
12 eleq1 2829 . . . . . . . . . . . . . 14 ((rank‘(𝐴𝐵)) = suc 𝐶 → ((rank‘(𝐴𝐵)) ∈ V ↔ suc 𝐶 ∈ V))
1311, 12mpbii 233 . . . . . . . . . . . . 13 ((rank‘(𝐴𝐵)) = suc 𝐶 → suc 𝐶 ∈ V)
14 sucexb 7824 . . . . . . . . . . . . 13 (𝐶 ∈ V ↔ suc 𝐶 ∈ V)
1513, 14sylibr 234 . . . . . . . . . . . 12 ((rank‘(𝐴𝐵)) = suc 𝐶𝐶 ∈ V)
16 nlimsucg 7863 . . . . . . . . . . . 12 (𝐶 ∈ V → ¬ Lim suc 𝐶)
1715, 16syl 17 . . . . . . . . . . 11 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim suc 𝐶)
18 limeq 6396 . . . . . . . . . . 11 ((rank‘(𝐴𝐵)) = suc 𝐶 → (Lim (rank‘(𝐴𝐵)) ↔ Lim suc 𝐶))
1917, 18mtbird 325 . . . . . . . . . 10 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴𝐵)))
20 rankxplim.1 . . . . . . . . . . 11 𝐴 ∈ V
21 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2220, 21rankxplim2 9920 . . . . . . . . . 10 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))
2319, 22nsyl 140 . . . . . . . . 9 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵)))
2420, 21xpex 7773 . . . . . . . . . . . . . 14 (𝐴 × 𝐵) ∈ V
2524rankeq0 9901 . . . . . . . . . . . . 13 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
2625necon3abii 2987 . . . . . . . . . . . 12 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
27 rankon 9835 . . . . . . . . . . . . . . . 16 (rank‘(𝐴 × 𝐵)) ∈ On
2827onordi 6495 . . . . . . . . . . . . . . 15 Ord (rank‘(𝐴 × 𝐵))
29 ordzsl 7866 . . . . . . . . . . . . . . 15 (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3028, 29mpbi 230 . . . . . . . . . . . . . 14 ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
31 3orass 1090 . . . . . . . . . . . . . 14 (((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))))
3230, 31mpbi 230 . . . . . . . . . . . . 13 ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3332ori 862 . . . . . . . . . . . 12 (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3426, 33sylbi 217 . . . . . . . . . . 11 ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3534ord 865 . . . . . . . . . 10 ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
3635con1d 145 . . . . . . . . 9 ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
3723, 36syl5com 31 . . . . . . . 8 ((rank‘(𝐴𝐵)) = suc 𝐶 → ((𝐴 × 𝐵) ≠ ∅ → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
38 nlimsucg 7863 . . . . . . . . . . . 12 (𝑥 ∈ V → ¬ Lim suc 𝑥)
3938elv 3485 . . . . . . . . . . 11 ¬ Lim suc 𝑥
40 limeq 6396 . . . . . . . . . . 11 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim suc 𝑥))
4139, 40mtbiri 327 . . . . . . . . . 10 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4241rexlimivw 3151 . . . . . . . . 9 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4320, 21rankxplim3 9921 . . . . . . . . 9 (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 × 𝐵)))
4442, 43sylnib 328 . . . . . . . 8 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4537, 44syl6com 37 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵))))
46 unixp0 6303 . . . . . . . . . . . 12 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
4724uniex 7761 . . . . . . . . . . . . 13 (𝐴 × 𝐵) ∈ V
4847rankeq0 9901 . . . . . . . . . . . 12 ( (𝐴 × 𝐵) = ∅ ↔ (rank‘ (𝐴 × 𝐵)) = ∅)
494eqeq1i 2742 . . . . . . . . . . . 12 ((rank‘ (𝐴 × 𝐵)) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
5046, 48, 493bitri 297 . . . . . . . . . . 11 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
5150necon3abii 2987 . . . . . . . . . 10 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
52 onuni 7808 . . . . . . . . . . . . . . 15 ((rank‘(𝐴 × 𝐵)) ∈ On → (rank‘(𝐴 × 𝐵)) ∈ On)
5327, 52ax-mp 5 . . . . . . . . . . . . . 14 (rank‘(𝐴 × 𝐵)) ∈ On
5453onordi 6495 . . . . . . . . . . . . 13 Ord (rank‘(𝐴 × 𝐵))
55 ordzsl 7866 . . . . . . . . . . . . 13 (Ord (rank‘(𝐴 × 𝐵)) ↔ ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
5654, 55mpbi 230 . . . . . . . . . . . 12 ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
57 3orass 1090 . . . . . . . . . . . 12 (( (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))) ↔ ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))))
5856, 57mpbi 230 . . . . . . . . . . 11 ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
5958ori 862 . . . . . . . . . 10 (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6051, 59sylbi 217 . . . . . . . . 9 ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6160ord 865 . . . . . . . 8 ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
6261con1d 145 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
6345, 62syld 47 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴𝐵)) = suc 𝐶 → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
6463impcom 407 . . . . 5 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
65 onsucuni2 7854 . . . . . . 7 (( (rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6653, 65mpan 690 . . . . . 6 ( (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6766rexlimivw 3151 . . . . 5 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6864, 67syl 17 . . . 4 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6910, 68eqtrd 2777 . . 3 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
70 suc11reg 9659 . . 3 (suc suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
7169, 70sylibr 234 . 2 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)))
7237imp 406 . . 3 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
73 onsucuni2 7854 . . . . 5 (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7427, 73mpan 690 . . . 4 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7574rexlimivw 3151 . . 3 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7672, 75syl 17 . 2 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7771, 76eqtr2d 2778 1 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848  w3o 1086   = wceq 1540  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  cun 3949  c0 4333   cuni 4907   × cxp 5683  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  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: (None)
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