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Theorem cfpwsdom 9259
Description: A corollary of Konig's Theorem konigth 9244. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
Hypothesis
Ref Expression
cfpwsdom.1 𝐵 ∈ V
Assertion
Ref Expression
cfpwsdom (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))

Proof of Theorem cfpwsdom
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6552 . . . . . . . . 9 (𝐵𝑚 (ℵ‘𝐴)) ∈ V
21cardid 9222 . . . . . . . 8 (card‘(𝐵𝑚 (ℵ‘𝐴))) ≈ (𝐵𝑚 (ℵ‘𝐴))
32ensymi 7866 . . . . . . 7 (𝐵𝑚 (ℵ‘𝐴)) ≈ (card‘(𝐵𝑚 (ℵ‘𝐴)))
4 fvex 6095 . . . . . . . . . . . . . 14 (ℵ‘𝐴) ∈ V
54canth2 7972 . . . . . . . . . . . . 13 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
64pw2en 7926 . . . . . . . . . . . . 13 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))
7 sdomentr 7953 . . . . . . . . . . . . 13 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)))
85, 6, 7mp2an 703 . . . . . . . . . . . 12 (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴))
9 mapdom1 7984 . . . . . . . . . . . 12 (2𝑜𝐵 → (2𝑜𝑚 (ℵ‘𝐴)) ≼ (𝐵𝑚 (ℵ‘𝐴)))
10 sdomdomtr 7952 . . . . . . . . . . . 12 (((ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)) ∧ (2𝑜𝑚 (ℵ‘𝐴)) ≼ (𝐵𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)))
118, 9, 10sylancr 693 . . . . . . . . . . 11 (2𝑜𝐵 → (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)))
12 ficard 9240 . . . . . . . . . . . . . . . . 17 ((𝐵𝑚 (ℵ‘𝐴)) ∈ V → ((𝐵𝑚 (ℵ‘𝐴)) ∈ Fin ↔ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω))
131, 12ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐵𝑚 (ℵ‘𝐴)) ∈ Fin ↔ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω)
14 isfinite 8406 . . . . . . . . . . . . . . . . 17 ((𝐵𝑚 (ℵ‘𝐴)) ∈ Fin ↔ (𝐵𝑚 (ℵ‘𝐴)) ≺ ω)
15 sdomdom 7843 . . . . . . . . . . . . . . . . 17 ((𝐵𝑚 (ℵ‘𝐴)) ≺ ω → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
1614, 15sylbi 205 . . . . . . . . . . . . . . . 16 ((𝐵𝑚 (ℵ‘𝐴)) ∈ Fin → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
1713, 16sylbir 223 . . . . . . . . . . . . . . 15 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
18 alephgeom 8762 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
19 alephon 8749 . . . . . . . . . . . . . . . . 17 (ℵ‘𝐴) ∈ On
20 ssdomg 7861 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
2119, 20ax-mp 5 . . . . . . . . . . . . . . . 16 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
2218, 21sylbi 205 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
23 domtr 7869 . . . . . . . . . . . . . . 15 (((𝐵𝑚 (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
2417, 22, 23syl2an 492 . . . . . . . . . . . . . 14 (((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
25 domnsym 7945 . . . . . . . . . . . . . 14 ((𝐵𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)))
2624, 25syl 17 . . . . . . . . . . . . 13 (((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → ¬ (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)))
2726expcom 449 . . . . . . . . . . . 12 (𝐴 ∈ On → ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → ¬ (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴))))
2827con2d 127 . . . . . . . . . . 11 (𝐴 ∈ On → ((ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)) → ¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω))
29 cardidm 8642 . . . . . . . . . . . 12 (card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴)))
30 iscard3 8773 . . . . . . . . . . . . 13 ((card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴))) ↔ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
31 elun 3711 . . . . . . . . . . . . 13 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ) ↔ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
32 df-or 383 . . . . . . . . . . . . 13 (((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ) ↔ (¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
3330, 31, 323bitri 284 . . . . . . . . . . . 12 ((card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴))) ↔ (¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
3429, 33mpbi 218 . . . . . . . . . . 11 (¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ)
3511, 28, 34syl56 35 . . . . . . . . . 10 (𝐴 ∈ On → (2𝑜𝐵 → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
36 alephfnon 8745 . . . . . . . . . . 11 ℵ Fn On
37 fvelrnb 6135 . . . . . . . . . . 11 (ℵ Fn On → ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴)))))
3836, 37ax-mp 5 . . . . . . . . . 10 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))))
3935, 38syl6ib 239 . . . . . . . . 9 (𝐴 ∈ On → (2𝑜𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴)))))
40 eqid 2606 . . . . . . . . . . . 12 (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦)))
4140pwcfsdom 9258 . . . . . . . . . . 11 (ℵ‘𝑥) ≺ ((ℵ‘𝑥) ↑𝑚 (cf‘(ℵ‘𝑥)))
42 id 22 . . . . . . . . . . . 12 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))))
43 fveq2 6085 . . . . . . . . . . . . 13 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
4442, 43oveq12d 6542 . . . . . . . . . . . 12 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → ((ℵ‘𝑥) ↑𝑚 (cf‘(ℵ‘𝑥))) = ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
4542, 44breq12d 4587 . . . . . . . . . . 11 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → ((ℵ‘𝑥) ≺ ((ℵ‘𝑥) ↑𝑚 (cf‘(ℵ‘𝑥))) ↔ (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
4641, 45mpbii 221 . . . . . . . . . 10 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
4746rexlimivw 3007 . . . . . . . . 9 (∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
4839, 47syl6 34 . . . . . . . 8 (𝐴 ∈ On → (2𝑜𝐵 → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
4948imp 443 . . . . . . 7 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
50 ensdomtr 7955 . . . . . . 7 (((𝐵𝑚 (ℵ‘𝐴)) ≈ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∧ (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))) → (𝐵𝑚 (ℵ‘𝐴)) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
513, 49, 50sylancr 693 . . . . . 6 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (𝐵𝑚 (ℵ‘𝐴)) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
52 fvex 6095 . . . . . . . . 9 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ∈ V
5352enref 7848 . . . . . . . 8 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))
54 mapen 7983 . . . . . . . 8 (((card‘(𝐵𝑚 (ℵ‘𝐴))) ≈ (𝐵𝑚 (ℵ‘𝐴)) ∧ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) → ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ ((𝐵𝑚 (ℵ‘𝐴)) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
552, 53, 54mp2an 703 . . . . . . 7 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ ((𝐵𝑚 (ℵ‘𝐴)) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
56 cfpwsdom.1 . . . . . . . 8 𝐵 ∈ V
57 mapxpen 7985 . . . . . . . 8 ((𝐵 ∈ V ∧ (ℵ‘𝐴) ∈ On ∧ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ∈ V) → ((𝐵𝑚 (ℵ‘𝐴)) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
5856, 19, 52, 57mp3an 1415 . . . . . . 7 ((𝐵𝑚 (ℵ‘𝐴)) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
5955, 58entri 7870 . . . . . 6 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
60 sdomentr 7953 . . . . . 6 (((𝐵𝑚 (ℵ‘𝐴)) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ∧ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))) → (𝐵𝑚 (ℵ‘𝐴)) ≺ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
6151, 59, 60sylancl 692 . . . . 5 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (𝐵𝑚 (ℵ‘𝐴)) ≺ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
624xpdom2 7914 . . . . . . . . . 10 ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6318biimpi 204 . . . . . . . . . . 11 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
64 infxpen 8694 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6519, 63, 64sylancr 693 . . . . . . . . . 10 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
66 domentr 7875 . . . . . . . . . 10 ((((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)) ∧ ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
6762, 65, 66syl2an 492 . . . . . . . . 9 (((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
68 nsuceq0 5705 . . . . . . . . . . 11 suc 1𝑜 ≠ ∅
69 dom0 7947 . . . . . . . . . . 11 (suc 1𝑜 ≼ ∅ ↔ suc 1𝑜 = ∅)
7068, 69nemtbir 2873 . . . . . . . . . 10 ¬ suc 1𝑜 ≼ ∅
71 df-2o 7422 . . . . . . . . . . . . . 14 2𝑜 = suc 1𝑜
7271breq1i 4581 . . . . . . . . . . . . 13 (2𝑜𝐵 ↔ suc 1𝑜𝐵)
73 breq2 4578 . . . . . . . . . . . . 13 (𝐵 = ∅ → (suc 1𝑜𝐵 ↔ suc 1𝑜 ≼ ∅))
7472, 73syl5bb 270 . . . . . . . . . . . 12 (𝐵 = ∅ → (2𝑜𝐵 ↔ suc 1𝑜 ≼ ∅))
7574biimpcd 237 . . . . . . . . . . 11 (2𝑜𝐵 → (𝐵 = ∅ → suc 1𝑜 ≼ ∅))
7675adantld 481 . . . . . . . . . 10 (2𝑜𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1𝑜 ≼ ∅))
7770, 76mtoi 188 . . . . . . . . 9 (2𝑜𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
78 mapdom2 7990 . . . . . . . . 9 ((((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ (ℵ‘𝐴) ∧ ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅)) → (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))) ≼ (𝐵𝑚 (ℵ‘𝐴)))
7967, 77, 78syl2an 492 . . . . . . . 8 ((((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2𝑜𝐵) → (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))) ≼ (𝐵𝑚 (ℵ‘𝐴)))
80 domnsym 7945 . . . . . . . 8 ((𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))) ≼ (𝐵𝑚 (ℵ‘𝐴)) → ¬ (𝐵𝑚 (ℵ‘𝐴)) ≺ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
8179, 80syl 17 . . . . . . 7 ((((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2𝑜𝐵) → ¬ (𝐵𝑚 (ℵ‘𝐴)) ≺ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
8281expl 645 . . . . . 6 ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ 2𝑜𝐵) → ¬ (𝐵𝑚 (ℵ‘𝐴)) ≺ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))))
8382com12 32 . . . . 5 ((𝐴 ∈ On ∧ 2𝑜𝐵) → ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ¬ (𝐵𝑚 (ℵ‘𝐴)) ≺ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))))
8461, 83mt2d 129 . . . 4 ((𝐴 ∈ On ∧ 2𝑜𝐵) → ¬ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
85 domtri 9231 . . . . . 6 (((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ∈ V ∧ (ℵ‘𝐴) ∈ V) → ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
8652, 4, 85mp2an 703 . . . . 5 ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
8786biimpri 216 . . . 4 (¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
8884, 87nsyl2 140 . . 3 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
8988ex 448 . 2 (𝐴 ∈ On → (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
90 fndm 5887 . . . . . 6 (ℵ Fn On → dom ℵ = On)
9136, 90ax-mp 5 . . . . 5 dom ℵ = On
9291eleq2i 2676 . . . 4 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
93 ndmfv 6110 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
9492, 93sylnbir 319 . . 3 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
95 1n0 7436 . . . . . 6 1𝑜 ≠ ∅
96 1onn 7580 . . . . . . . 8 1𝑜 ∈ ω
9796elexi 3182 . . . . . . 7 1𝑜 ∈ V
98970sdom 7950 . . . . . 6 (∅ ≺ 1𝑜 ↔ 1𝑜 ≠ ∅)
9995, 98mpbir 219 . . . . 5 ∅ ≺ 1𝑜
100 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
101 oveq2 6532 . . . . . . . . . . 11 ((ℵ‘𝐴) = ∅ → (𝐵𝑚 (ℵ‘𝐴)) = (𝐵𝑚 ∅))
102 map0e 7755 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝐵𝑚 ∅) = 1𝑜)
10356, 102ax-mp 5 . . . . . . . . . . 11 (𝐵𝑚 ∅) = 1𝑜
104101, 103syl6eq 2656 . . . . . . . . . 10 ((ℵ‘𝐴) = ∅ → (𝐵𝑚 (ℵ‘𝐴)) = 1𝑜)
105104fveq2d 6089 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (card‘(𝐵𝑚 (ℵ‘𝐴))) = (card‘1𝑜))
106 cardnn 8646 . . . . . . . . . 10 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
10796, 106ax-mp 5 . . . . . . . . 9 (card‘1𝑜) = 1𝑜
108105, 107syl6eq 2656 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (card‘(𝐵𝑚 (ℵ‘𝐴))) = 1𝑜)
109108fveq2d 6089 . . . . . . 7 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (cf‘1𝑜))
110 df-1o 7421 . . . . . . . . 9 1𝑜 = suc ∅
111110fveq2i 6088 . . . . . . . 8 (cf‘1𝑜) = (cf‘suc ∅)
112 0elon 5678 . . . . . . . . 9 ∅ ∈ On
113 cfsuc 8936 . . . . . . . . 9 (∅ ∈ On → (cf‘suc ∅) = 1𝑜)
114112, 113ax-mp 5 . . . . . . . 8 (cf‘suc ∅) = 1𝑜
115111, 114eqtri 2628 . . . . . . 7 (cf‘1𝑜) = 1𝑜
116109, 115syl6eq 2656 . . . . . 6 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = 1𝑜)
117100, 116breq12d 4587 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ↔ ∅ ≺ 1𝑜))
11899, 117mpbiri 246 . . . 4 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
119118a1d 25 . . 3 ((ℵ‘𝐴) = ∅ → (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
12094, 119syl 17 . 2 𝐴 ∈ On → (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
12189, 120pm2.61i 174 1 (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2776  wrex 2893  Vcvv 3169  cun 3534  wss 3536  c0 3870  𝒫 cpw 4104   class class class wbr 4574  cmpt 4634   × cxp 5023  dom cdm 5025  ran crn 5026  Oncon0 5623  suc csuc 5625   Fn wfn 5782  cfv 5787  (class class class)co 6524  ωcom 6931  1𝑜c1o 7414  2𝑜c2o 7415  𝑚 cmap 7718  cen 7812  cdom 7813  csdm 7814  Fincfn 7815  harchar 8318  cardccrd 8618  cale 8619  cfccf 8620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-ac2 9142
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 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-iin 4449  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-smo 7304  df-recs 7329  df-rdg 7367  df-1o 7421  df-2o 7422  df-oadd 7425  df-er 7603  df-map 7720  df-ixp 7769  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-oi 8272  df-har 8320  df-card 8622  df-aleph 8623  df-cf 8624  df-acn 8625  df-ac 8796
This theorem is referenced by:  alephom  9260
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