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Theorem rankxpsuc 8601
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 8598 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 rankuni 8582 . . . . . . . 8 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
2 rankuni 8582 . . . . . . . . 9 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
32unieqi 4371 . . . . . . . 8 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
41, 3eqtri 2627 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
5 unixp 5567 . . . . . . . 8 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
65fveq2d 6088 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
74, 6syl5reqr 2654 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
8 suc11reg 8372 . . . . . 6 (suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)) ↔ (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
97, 8sylibr 222 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)))
109adantl 480 . . . 4 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)))
11 fvex 6094 . . . . . . . . . . . . . 14 (rank‘(𝐴𝐵)) ∈ V
12 eleq1 2671 . . . . . . . . . . . . . 14 ((rank‘(𝐴𝐵)) = suc 𝐶 → ((rank‘(𝐴𝐵)) ∈ V ↔ suc 𝐶 ∈ V))
1311, 12mpbii 221 . . . . . . . . . . . . 13 ((rank‘(𝐴𝐵)) = suc 𝐶 → suc 𝐶 ∈ V)
14 sucexb 6874 . . . . . . . . . . . . 13 (𝐶 ∈ V ↔ suc 𝐶 ∈ V)
1513, 14sylibr 222 . . . . . . . . . . . 12 ((rank‘(𝐴𝐵)) = suc 𝐶𝐶 ∈ V)
16 nlimsucg 6907 . . . . . . . . . . . 12 (𝐶 ∈ V → ¬ Lim suc 𝐶)
1715, 16syl 17 . . . . . . . . . . 11 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim suc 𝐶)
18 limeq 5634 . . . . . . . . . . 11 ((rank‘(𝐴𝐵)) = suc 𝐶 → (Lim (rank‘(𝐴𝐵)) ↔ Lim suc 𝐶))
1917, 18mtbird 313 . . . . . . . . . 10 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴𝐵)))
20 rankxplim.1 . . . . . . . . . . 11 𝐴 ∈ V
21 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2220, 21rankxplim2 8599 . . . . . . . . . 10 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))
2319, 22nsyl 133 . . . . . . . . 9 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵)))
2420, 21xpex 6833 . . . . . . . . . . . . . 14 (𝐴 × 𝐵) ∈ V
2524rankeq0 8580 . . . . . . . . . . . . 13 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
2625necon3abii 2823 . . . . . . . . . . . 12 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
27 rankon 8514 . . . . . . . . . . . . . . . 16 (rank‘(𝐴 × 𝐵)) ∈ On
2827onordi 5731 . . . . . . . . . . . . . . 15 Ord (rank‘(𝐴 × 𝐵))
29 ordzsl 6910 . . . . . . . . . . . . . . 15 (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3028, 29mpbi 218 . . . . . . . . . . . . . 14 ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
31 3orass 1033 . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . 13 ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3332ori 388 . . . . . . . . . . . 12 (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3426, 33sylbi 205 . . . . . . . . . . 11 ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3534ord 390 . . . . . . . . . 10 ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
3635con1d 137 . . . . . . . . 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 vex 3171 . . . . . . . . . . . 12 𝑥 ∈ V
39 nlimsucg 6907 . . . . . . . . . . . 12 (𝑥 ∈ V → ¬ Lim suc 𝑥)
4038, 39ax-mp 5 . . . . . . . . . . 11 ¬ Lim suc 𝑥
41 limeq 5634 . . . . . . . . . . 11 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim suc 𝑥))
4240, 41mtbiri 315 . . . . . . . . . 10 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4342rexlimivw 3006 . . . . . . . . 9 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4420, 21rankxplim3 8600 . . . . . . . . 9 (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 × 𝐵)))
4543, 44sylnib 316 . . . . . . . 8 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4637, 45syl6com 36 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵))))
47 unixp0 5568 . . . . . . . . . . . 12 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
4824uniex 6824 . . . . . . . . . . . . 13 (𝐴 × 𝐵) ∈ V
4948rankeq0 8580 . . . . . . . . . . . 12 ( (𝐴 × 𝐵) = ∅ ↔ (rank‘ (𝐴 × 𝐵)) = ∅)
502eqeq1i 2610 . . . . . . . . . . . 12 ((rank‘ (𝐴 × 𝐵)) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
5147, 49, 503bitri 284 . . . . . . . . . . 11 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
5251necon3abii 2823 . . . . . . . . . 10 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
53 onuni 6858 . . . . . . . . . . . . . . 15 ((rank‘(𝐴 × 𝐵)) ∈ On → (rank‘(𝐴 × 𝐵)) ∈ On)
5427, 53ax-mp 5 . . . . . . . . . . . . . 14 (rank‘(𝐴 × 𝐵)) ∈ On
5554onordi 5731 . . . . . . . . . . . . 13 Ord (rank‘(𝐴 × 𝐵))
56 ordzsl 6910 . . . . . . . . . . . . 13 (Ord (rank‘(𝐴 × 𝐵)) ↔ ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
5755, 56mpbi 218 . . . . . . . . . . . 12 ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
58 3orass 1033 . . . . . . . . . . . 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)))))
5957, 58mpbi 218 . . . . . . . . . . 11 ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6059ori 388 . . . . . . . . . 10 (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6152, 60sylbi 205 . . . . . . . . 9 ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6261ord 390 . . . . . . . 8 ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
6362con1d 137 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
6446, 63syld 45 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴𝐵)) = suc 𝐶 → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
6564impcom 444 . . . . 5 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
66 onsucuni2 6899 . . . . . . 7 (( (rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6754, 66mpan 701 . . . . . 6 ( (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6867rexlimivw 3006 . . . . 5 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6965, 68syl 17 . . . 4 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7010, 69eqtrd 2639 . . 3 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
71 suc11reg 8372 . . 3 (suc suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
7270, 71sylibr 222 . 2 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)))
7337imp 443 . . 3 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
74 onsucuni2 6899 . . . . 5 (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7527, 74mpan 701 . . . 4 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7675rexlimivw 3006 . . 3 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7773, 76syl 17 . 2 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7872, 77eqtr2d 2640 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  wo 381  wa 382  w3o 1029   = wceq 1474  wcel 1975  wne 2775  wrex 2892  Vcvv 3168  cun 3533  c0 3869   cuni 4362   × cxp 5022  Ord word 5621  Oncon0 5622  Lim wlim 5623  suc csuc 5624  cfv 5786  rankcrnk 8482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-reg 8353  ax-inf2 8394
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-r1 8483  df-rank 8484
This theorem is referenced by: (None)
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