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Theorem cfpwsdom 9391
Description: A corollary of Konig's Theorem konigth 9376. 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 6663 . . . . . . . . 9 (𝐵𝑚 (ℵ‘𝐴)) ∈ V
21cardid 9354 . . . . . . . 8 (card‘(𝐵𝑚 (ℵ‘𝐴))) ≈ (𝐵𝑚 (ℵ‘𝐴))
32ensymi 7991 . . . . . . 7 (𝐵𝑚 (ℵ‘𝐴)) ≈ (card‘(𝐵𝑚 (ℵ‘𝐴)))
4 fvex 6188 . . . . . . . . . . . . . 14 (ℵ‘𝐴) ∈ V
54canth2 8098 . . . . . . . . . . . . 13 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
64pw2en 8052 . . . . . . . . . . . . 13 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))
7 sdomentr 8079 . . . . . . . . . . . . 13 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)))
85, 6, 7mp2an 707 . . . . . . . . . . . 12 (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴))
9 mapdom1 8110 . . . . . . . . . . . 12 (2𝑜𝐵 → (2𝑜𝑚 (ℵ‘𝐴)) ≼ (𝐵𝑚 (ℵ‘𝐴)))
10 sdomdomtr 8078 . . . . . . . . . . . 12 (((ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)) ∧ (2𝑜𝑚 (ℵ‘𝐴)) ≼ (𝐵𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)))
118, 9, 10sylancr 694 . . . . . . . . . . 11 (2𝑜𝐵 → (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)))
12 ficard 9372 . . . . . . . . . . . . . . . . 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 8534 . . . . . . . . . . . . . . . . 17 ((𝐵𝑚 (ℵ‘𝐴)) ∈ Fin ↔ (𝐵𝑚 (ℵ‘𝐴)) ≺ ω)
15 sdomdom 7968 . . . . . . . . . . . . . . . . 17 ((𝐵𝑚 (ℵ‘𝐴)) ≺ ω → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
1614, 15sylbi 207 . . . . . . . . . . . . . . . 16 ((𝐵𝑚 (ℵ‘𝐴)) ∈ Fin → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
1713, 16sylbir 225 . . . . . . . . . . . . . . 15 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
18 alephgeom 8890 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
19 alephon 8877 . . . . . . . . . . . . . . . . 17 (ℵ‘𝐴) ∈ On
20 ssdomg 7986 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
2119, 20ax-mp 5 . . . . . . . . . . . . . . . 16 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
2218, 21sylbi 207 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
23 domtr 7994 . . . . . . . . . . . . . . 15 (((𝐵𝑚 (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
2417, 22, 23syl2an 494 . . . . . . . . . . . . . 14 (((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
25 domnsym 8071 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . 12 (𝐴 ∈ On → ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → ¬ (ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴))))
2827con2d 129 . . . . . . . . . . 11 (𝐴 ∈ On → ((ℵ‘𝐴) ≺ (𝐵𝑚 (ℵ‘𝐴)) → ¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω))
29 cardidm 8770 . . . . . . . . . . . 12 (card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴)))
30 iscard3 8901 . . . . . . . . . . . . 13 ((card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴))) ↔ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
31 elun 3745 . . . . . . . . . . . . 13 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ) ↔ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
32 df-or 385 . . . . . . . . . . . . 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 286 . . . . . . . . . . . 12 ((card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴))) ↔ (¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
3429, 33mpbi 220 . . . . . . . . . . 11 (¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ)
3511, 28, 34syl56 36 . . . . . . . . . 10 (𝐴 ∈ On → (2𝑜𝐵 → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
36 alephfnon 8873 . . . . . . . . . . 11 ℵ Fn On
37 fvelrnb 6230 . . . . . . . . . . 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 241 . . . . . . . . 9 (𝐴 ∈ On → (2𝑜𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴)))))
40 eqid 2620 . . . . . . . . . . . 12 (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦)))
4140pwcfsdom 9390 . . . . . . . . . . 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 6178 . . . . . . . . . . . . 13 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
4442, 43oveq12d 6653 . . . . . . . . . . . 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 4657 . . . . . . . . . . 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 223 . . . . . . . . . 10 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
4746rexlimivw 3025 . . . . . . . . 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 35 . . . . . . . 8 (𝐴 ∈ On → (2𝑜𝐵 → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
4948imp 445 . . . . . . 7 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
50 ensdomtr 8081 . . . . . . 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 694 . . . . . 6 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (𝐵𝑚 (ℵ‘𝐴)) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
52 fvex 6188 . . . . . . . . 9 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ∈ V
5352enref 7973 . . . . . . . 8 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))
54 mapen 8109 . . . . . . . 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 707 . . . . . . 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 8111 . . . . . . . 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 1422 . . . . . . 7 ((𝐵𝑚 (ℵ‘𝐴)) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
5955, 58entri 7995 . . . . . 6 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
60 sdomentr 8079 . . . . . 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 693 . . . . 5 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (𝐵𝑚 (ℵ‘𝐴)) ≺ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))))
624xpdom2 8040 . . . . . . . . . 10 ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6318biimpi 206 . . . . . . . . . . 11 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
64 infxpen 8822 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6519, 63, 64sylancr 694 . . . . . . . . . 10 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
66 domentr 8000 . . . . . . . . . 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 494 . . . . . . . . 9 (((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
68 nsuceq0 5793 . . . . . . . . . . 11 suc 1𝑜 ≠ ∅
69 dom0 8073 . . . . . . . . . . 11 (suc 1𝑜 ≼ ∅ ↔ suc 1𝑜 = ∅)
7068, 69nemtbir 2886 . . . . . . . . . 10 ¬ suc 1𝑜 ≼ ∅
71 df-2o 7546 . . . . . . . . . . . . . 14 2𝑜 = suc 1𝑜
7271breq1i 4651 . . . . . . . . . . . . 13 (2𝑜𝐵 ↔ suc 1𝑜𝐵)
73 breq2 4648 . . . . . . . . . . . . 13 (𝐵 = ∅ → (suc 1𝑜𝐵 ↔ suc 1𝑜 ≼ ∅))
7472, 73syl5bb 272 . . . . . . . . . . . 12 (𝐵 = ∅ → (2𝑜𝐵 ↔ suc 1𝑜 ≼ ∅))
7574biimpcd 239 . . . . . . . . . . 11 (2𝑜𝐵 → (𝐵 = ∅ → suc 1𝑜 ≼ ∅))
7675adantld 483 . . . . . . . . . 10 (2𝑜𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1𝑜 ≼ ∅))
7770, 76mtoi 190 . . . . . . . . 9 (2𝑜𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
78 mapdom2 8116 . . . . . . . . 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 494 . . . . . . . 8 ((((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2𝑜𝐵) → (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))) ≼ (𝐵𝑚 (ℵ‘𝐴)))
80 domnsym 8071 . . . . . . . 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 647 . . . . . 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 131 . . . 4 ((𝐴 ∈ On ∧ 2𝑜𝐵) → ¬ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
85 domtri 9363 . . . . . 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 707 . . . . 5 ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
8786biimpri 218 . . . 4 (¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
8884, 87nsyl2 142 . . 3 ((𝐴 ∈ On ∧ 2𝑜𝐵) → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
8988ex 450 . 2 (𝐴 ∈ On → (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
90 fndm 5978 . . . . . 6 (ℵ Fn On → dom ℵ = On)
9136, 90ax-mp 5 . . . . 5 dom ℵ = On
9291eleq2i 2691 . . . 4 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
93 ndmfv 6205 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
9492, 93sylnbir 321 . . 3 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
95 1n0 7560 . . . . . 6 1𝑜 ≠ ∅
96 1onn 7704 . . . . . . . 8 1𝑜 ∈ ω
9796elexi 3208 . . . . . . 7 1𝑜 ∈ V
98970sdom 8076 . . . . . 6 (∅ ≺ 1𝑜 ↔ 1𝑜 ≠ ∅)
9995, 98mpbir 221 . . . . 5 ∅ ≺ 1𝑜
100 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
101 oveq2 6643 . . . . . . . . . . 11 ((ℵ‘𝐴) = ∅ → (𝐵𝑚 (ℵ‘𝐴)) = (𝐵𝑚 ∅))
102 map0e 7880 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝐵𝑚 ∅) = 1𝑜)
10356, 102ax-mp 5 . . . . . . . . . . 11 (𝐵𝑚 ∅) = 1𝑜
104101, 103syl6eq 2670 . . . . . . . . . 10 ((ℵ‘𝐴) = ∅ → (𝐵𝑚 (ℵ‘𝐴)) = 1𝑜)
105104fveq2d 6182 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (card‘(𝐵𝑚 (ℵ‘𝐴))) = (card‘1𝑜))
106 cardnn 8774 . . . . . . . . . 10 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
10796, 106ax-mp 5 . . . . . . . . 9 (card‘1𝑜) = 1𝑜
108105, 107syl6eq 2670 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (card‘(𝐵𝑚 (ℵ‘𝐴))) = 1𝑜)
109108fveq2d 6182 . . . . . . 7 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (cf‘1𝑜))
110 df-1o 7545 . . . . . . . . 9 1𝑜 = suc ∅
111110fveq2i 6181 . . . . . . . 8 (cf‘1𝑜) = (cf‘suc ∅)
112 0elon 5766 . . . . . . . . 9 ∅ ∈ On
113 cfsuc 9064 . . . . . . . . 9 (∅ ∈ On → (cf‘suc ∅) = 1𝑜)
114112, 113ax-mp 5 . . . . . . . 8 (cf‘suc ∅) = 1𝑜
115111, 114eqtri 2642 . . . . . . 7 (cf‘1𝑜) = 1𝑜
116109, 115syl6eq 2670 . . . . . 6 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = 1𝑜)
117100, 116breq12d 4657 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ↔ ∅ ≺ 1𝑜))
11899, 117mpbiri 248 . . . 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 176 1 (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1481  wcel 1988  wne 2791  wrex 2910  Vcvv 3195  cun 3565  wss 3567  c0 3907  𝒫 cpw 4149   class class class wbr 4644  cmpt 4720   × cxp 5102  dom cdm 5104  ran crn 5105  Oncon0 5711  suc csuc 5713   Fn wfn 5871  cfv 5876  (class class class)co 6635  ωcom 7050  1𝑜c1o 7538  2𝑜c2o 7539  𝑚 cmap 7842  cen 7937  cdom 7938  csdm 7939  Fincfn 7940  harchar 8446  cardccrd 8746  cale 8747  cfccf 8748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-smo 7428  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750  df-aleph 8751  df-cf 8752  df-acn 8753  df-ac 8924
This theorem is referenced by:  alephom  9392
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