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Theorem rankxpsuc 8628
 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 8625 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 8609 . . . . . . . 8 (rank‘ (𝐴 × 𝐵)) = (rank‘ (𝐴 × 𝐵))
2 rankuni 8609 . . . . . . . . 9 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
32unieqi 4381 . . . . . . . 8 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
41, 3eqtri 2632 . . . . . . 7 (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵))
5 unixp 5585 . . . . . . . 8 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
65fveq2d 6107 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → (rank‘ (𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
74, 6syl5reqr 2659 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
8 suc11reg 8399 . . . . . 6 (suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)) ↔ (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
97, 8sylibr 223 . . . . 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 6113 . . . . . . . . . . . . . 14 (rank‘(𝐴𝐵)) ∈ V
12 eleq1 2676 . . . . . . . . . . . . . 14 ((rank‘(𝐴𝐵)) = suc 𝐶 → ((rank‘(𝐴𝐵)) ∈ V ↔ suc 𝐶 ∈ V))
1311, 12mpbii 222 . . . . . . . . . . . . 13 ((rank‘(𝐴𝐵)) = suc 𝐶 → suc 𝐶 ∈ V)
14 sucexb 6901 . . . . . . . . . . . . 13 (𝐶 ∈ V ↔ suc 𝐶 ∈ V)
1513, 14sylibr 223 . . . . . . . . . . . 12 ((rank‘(𝐴𝐵)) = suc 𝐶𝐶 ∈ V)
16 nlimsucg 6934 . . . . . . . . . . . 12 (𝐶 ∈ V → ¬ Lim suc 𝐶)
1715, 16syl 17 . . . . . . . . . . 11 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim suc 𝐶)
18 limeq 5652 . . . . . . . . . . 11 ((rank‘(𝐴𝐵)) = suc 𝐶 → (Lim (rank‘(𝐴𝐵)) ↔ Lim suc 𝐶))
1917, 18mtbird 314 . . . . . . . . . 10 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴𝐵)))
20 rankxplim.1 . . . . . . . . . . 11 𝐴 ∈ V
21 rankxplim.2 . . . . . . . . . . 11 𝐵 ∈ V
2220, 21rankxplim2 8626 . . . . . . . . . 10 (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))
2319, 22nsyl 134 . . . . . . . . 9 ((rank‘(𝐴𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵)))
2420, 21xpex 6860 . . . . . . . . . . . . . 14 (𝐴 × 𝐵) ∈ V
2524rankeq0 8607 . . . . . . . . . . . . 13 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
2625necon3abii 2828 . . . . . . . . . . . 12 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
27 rankon 8541 . . . . . . . . . . . . . . . 16 (rank‘(𝐴 × 𝐵)) ∈ On
2827onordi 5749 . . . . . . . . . . . . . . 15 Ord (rank‘(𝐴 × 𝐵))
29 ordzsl 6937 . . . . . . . . . . . . . . 15 (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3028, 29mpbi 219 . . . . . . . . . . . . . 14 ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
31 3orass 1034 . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . 13 ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3332ori 389 . . . . . . . . . . . 12 (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3426, 33sylbi 206 . . . . . . . . . . 11 ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
3534ord 391 . . . . . . . . . 10 ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
3635con1d 138 . . . . . . . . 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 3176 . . . . . . . . . . . 12 𝑥 ∈ V
39 nlimsucg 6934 . . . . . . . . . . . 12 (𝑥 ∈ V → ¬ Lim suc 𝑥)
4038, 39ax-mp 5 . . . . . . . . . . 11 ¬ Lim suc 𝑥
41 limeq 5652 . . . . . . . . . . 11 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim suc 𝑥))
4240, 41mtbiri 316 . . . . . . . . . 10 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4342rexlimivw 3011 . . . . . . . . 9 (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
4420, 21rankxplim3 8627 . . . . . . . . 9 (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 × 𝐵)))
4543, 44sylnib 317 . . . . . . . 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 5586 . . . . . . . . . . . 12 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
4824uniex 6851 . . . . . . . . . . . . 13 (𝐴 × 𝐵) ∈ V
4948rankeq0 8607 . . . . . . . . . . . 12 ( (𝐴 × 𝐵) = ∅ ↔ (rank‘ (𝐴 × 𝐵)) = ∅)
502eqeq1i 2615 . . . . . . . . . . . 12 ((rank‘ (𝐴 × 𝐵)) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
5147, 49, 503bitri 285 . . . . . . . . . . 11 ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
5251necon3abii 2828 . . . . . . . . . 10 ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
53 onuni 6885 . . . . . . . . . . . . . . 15 ((rank‘(𝐴 × 𝐵)) ∈ On → (rank‘(𝐴 × 𝐵)) ∈ On)
5427, 53ax-mp 5 . . . . . . . . . . . . . 14 (rank‘(𝐴 × 𝐵)) ∈ On
5554onordi 5749 . . . . . . . . . . . . 13 Ord (rank‘(𝐴 × 𝐵))
56 ordzsl 6937 . . . . . . . . . . . . 13 (Ord (rank‘(𝐴 × 𝐵)) ↔ ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
5755, 56mpbi 219 . . . . . . . . . . . 12 ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
58 3orass 1034 . . . . . . . . . . . 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 219 . . . . . . . . . . 11 ( (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6059ori 389 . . . . . . . . . 10 (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6152, 60sylbi 206 . . . . . . . . 9 ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
6261ord 391 . . . . . . . 8 ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
6362con1d 138 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
6446, 63syld 46 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴𝐵)) = suc 𝐶 → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
6564impcom 445 . . . . 5 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
66 onsucuni2 6926 . . . . . . 7 (( (rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6754, 66mpan 702 . . . . . 6 ( (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
6867rexlimivw 3011 . . . . 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 2644 . . 3 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
71 suc11reg 8399 . . 3 (suc suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴𝐵)) = (rank‘(𝐴 × 𝐵)))
7270, 71sylibr 223 . 2 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc suc (rank‘(𝐴𝐵)) = suc (rank‘(𝐴 × 𝐵)))
7337imp 444 . . 3 (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
74 onsucuni2 6926 . . . . 5 (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7527, 74mpan 702 . . . 4 ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
7675rexlimivw 3011 . . 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 2645 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 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∪ cun 3538  ∅c0 3874  ∪ cuni 4372   × cxp 5036  Ord word 5639  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: (None)
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