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Theorem cfpwsdom 10621
Description: A corollary of Konig's Theorem konigth 10606. 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 7463 . . . . . . . . 9 (𝐵m (ℵ‘𝐴)) ∈ V
21cardid 10584 . . . . . . . 8 (card‘(𝐵m (ℵ‘𝐴))) ≈ (𝐵m (ℵ‘𝐴))
32ensymi 9042 . . . . . . 7 (𝐵m (ℵ‘𝐴)) ≈ (card‘(𝐵m (ℵ‘𝐴)))
4 fvex 6919 . . . . . . . . . . . . . 14 (ℵ‘𝐴) ∈ V
54canth2 9168 . . . . . . . . . . . . 13 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
64pw2en 9117 . . . . . . . . . . . . 13 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))
7 sdomentr 9149 . . . . . . . . . . . . 13 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)))
85, 6, 7mp2an 692 . . . . . . . . . . . 12 (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴))
9 mapdom1 9180 . . . . . . . . . . . 12 (2o𝐵 → (2om (ℵ‘𝐴)) ≼ (𝐵m (ℵ‘𝐴)))
10 sdomdomtr 9148 . . . . . . . . . . . 12 (((ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)) ∧ (2om (ℵ‘𝐴)) ≼ (𝐵m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
118, 9, 10sylancr 587 . . . . . . . . . . 11 (2o𝐵 → (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
12 ficard 10602 . . . . . . . . . . . . . . . . 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 fict 9690 . . . . . . . . . . . . . . . 16 ((𝐵m (ℵ‘𝐴)) ∈ Fin → (𝐵m (ℵ‘𝐴)) ≼ ω)
1513, 14sylbir 235 . . . . . . . . . . . . . . 15 ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (𝐵m (ℵ‘𝐴)) ≼ ω)
16 alephgeom 10119 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 alephon 10106 . . . . . . . . . . . . . . . . 17 (ℵ‘𝐴) ∈ On
18 ssdomg 9038 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
1917, 18ax-mp 5 . . . . . . . . . . . . . . . 16 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
2016, 19sylbi 217 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21 domtr 9045 . . . . . . . . . . . . . . 15 (((𝐵m (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
2215, 20, 21syl2an 596 . . . . . . . . . . . . . 14 (((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
23 domnsym 9137 . . . . . . . . . . . . . 14 ((𝐵m (ℵ‘𝐴)) ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
2422, 23syl 17 . . . . . . . . . . . . 13 (((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → ¬ (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
2524expcom 413 . . . . . . . . . . . 12 (𝐴 ∈ On → ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω → ¬ (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴))))
2625con2d 134 . . . . . . . . . . 11 (𝐴 ∈ On → ((ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)) → ¬ (card‘(𝐵m (ℵ‘𝐴))) ∈ ω))
27 cardidm 9996 . . . . . . . . . . . 12 (card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴)))
28 iscard3 10130 . . . . . . . . . . . . 13 ((card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴))) ↔ (card‘(𝐵m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29 elun 4162 . . . . . . . . . . . . 13 ((card‘(𝐵m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ) ↔ ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
30 df-or 848 . . . . . . . . . . . . 13 (((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ) ↔ (¬ (card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
3128, 29, 303bitri 297 . . . . . . . . . . . 12 ((card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴))) ↔ (¬ (card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
3227, 31mpbi 230 . . . . . . . . . . 11 (¬ (card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ)
3311, 26, 32syl56 36 . . . . . . . . . 10 (𝐴 ∈ On → (2o𝐵 → (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
34 alephfnon 10102 . . . . . . . . . . 11 ℵ Fn On
35 fvelrnb 6968 . . . . . . . . . . 11 (ℵ Fn On → ((card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴)))))
3634, 35ax-mp 5 . . . . . . . . . 10 ((card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))))
3733, 36imbitrdi 251 . . . . . . . . 9 (𝐴 ∈ On → (2o𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴)))))
38 eqid 2734 . . . . . . . . . . . 12 (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦)))
3938pwcfsdom 10620 . . . . . . . . . . 11 (ℵ‘𝑥) ≺ ((ℵ‘𝑥) ↑m (cf‘(ℵ‘𝑥)))
40 id 22 . . . . . . . . . . . 12 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))))
41 fveq2 6906 . . . . . . . . . . . . 13 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
4240, 41oveq12d 7448 . . . . . . . . . . . 12 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → ((ℵ‘𝑥) ↑m (cf‘(ℵ‘𝑥))) = ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
4340, 42breq12d 5160 . . . . . . . . . . 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)))))))
4439, 43mpbii 233 . . . . . . . . . 10 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
4544rexlimivw 3148 . . . . . . . . 9 (∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
4637, 45syl6 35 . . . . . . . 8 (𝐴 ∈ On → (2o𝐵 → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴)))))))
4746imp 406 . . . . . . 7 ((𝐴 ∈ On ∧ 2o𝐵) → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
48 ensdomtr 9151 . . . . . . 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))))))
493, 47, 48sylancr 587 . . . . . 6 ((𝐴 ∈ On ∧ 2o𝐵) → (𝐵m (ℵ‘𝐴)) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
50 fvex 6919 . . . . . . . . 9 (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ∈ V
5150enref 9023 . . . . . . . 8 (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵m (ℵ‘𝐴))))
52 mapen 9179 . . . . . . . 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))))))
532, 51, 52mp2an 692 . . . . . . 7 ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≈ ((𝐵m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
54 cfpwsdom.1 . . . . . . . 8 𝐵 ∈ V
55 mapxpen 9181 . . . . . . . 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)))))))
5654, 17, 50, 55mp3an 1460 . . . . . . 7 ((𝐵m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≈ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
5753, 56entri 9046 . . . . . 6 ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≈ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
58 sdomentr 9149 . . . . . 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)))))))
5949, 57, 58sylancl 586 . . . . 5 ((𝐴 ∈ On ∧ 2o𝐵) → (𝐵m (ℵ‘𝐴)) ≺ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))))
604xpdom2 9105 . . . . . . . . . 10 ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6116biimpi 216 . . . . . . . . . . 11 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62 infxpen 10051 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6317, 61, 62sylancr 587 . . . . . . . . . 10 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
64 domentr 9051 . . . . . . . . . 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))
6560, 63, 64syl2an 596 . . . . . . . . 9 (((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
66 nsuceq0 6468 . . . . . . . . . . 11 suc 1o ≠ ∅
67 dom0 9140 . . . . . . . . . . 11 (suc 1o ≼ ∅ ↔ suc 1o = ∅)
6866, 67nemtbir 3035 . . . . . . . . . 10 ¬ suc 1o ≼ ∅
69 df-2o 8505 . . . . . . . . . . . . . 14 2o = suc 1o
7069breq1i 5154 . . . . . . . . . . . . 13 (2o𝐵 ↔ suc 1o𝐵)
71 breq2 5151 . . . . . . . . . . . . 13 (𝐵 = ∅ → (suc 1o𝐵 ↔ suc 1o ≼ ∅))
7270, 71bitrid 283 . . . . . . . . . . . 12 (𝐵 = ∅ → (2o𝐵 ↔ suc 1o ≼ ∅))
7372biimpcd 249 . . . . . . . . . . 11 (2o𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
7473adantld 490 . . . . . . . . . 10 (2o𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
7568, 74mtoi 199 . . . . . . . . 9 (2o𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76 mapdom2 9186 . . . . . . . . 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)))
7765, 75, 76syl2an 596 . . . . . . . 8 ((((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2o𝐵) → (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))) ≼ (𝐵m (ℵ‘𝐴)))
78 domnsym 9137 . . . . . . . 8 ((𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))) ≼ (𝐵m (ℵ‘𝐴)) → ¬ (𝐵m (ℵ‘𝐴)) ≺ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))))
7977, 78syl 17 . . . . . . 7 ((((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2o𝐵) → ¬ (𝐵m (ℵ‘𝐴)) ≺ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))))
8079expl 457 . . . . . 6 ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ 2o𝐵) → ¬ (𝐵m (ℵ‘𝐴)) ≺ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))))
8180com12 32 . . . . 5 ((𝐴 ∈ On ∧ 2o𝐵) → ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ¬ (𝐵m (ℵ‘𝐴)) ≺ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))))
8259, 81mt2d 136 . . . 4 ((𝐴 ∈ On ∧ 2o𝐵) → ¬ (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
83 domtri 10593 . . . . . 6 (((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ∈ V ∧ (ℵ‘𝐴) ∈ V) → ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
8450, 4, 83mp2an 692 . . . . 5 ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
8584biimpri 228 . . . 4 (¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))) → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
8682, 85nsyl2 141 . . 3 ((𝐴 ∈ On ∧ 2o𝐵) → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
8786ex 412 . 2 (𝐴 ∈ On → (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
88 fndm 6671 . . . . . 6 (ℵ Fn On → dom ℵ = On)
8934, 88ax-mp 5 . . . . 5 dom ℵ = On
9089eleq2i 2830 . . . 4 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91 ndmfv 6941 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
9290, 91sylnbir 331 . . 3 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93 1n0 8524 . . . . . 6 1o ≠ ∅
94 1oex 8514 . . . . . . 7 1o ∈ V
95940sdom 9145 . . . . . 6 (∅ ≺ 1o ↔ 1o ≠ ∅)
9693, 95mpbir 231 . . . . 5 ∅ ≺ 1o
97 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98 oveq2 7438 . . . . . . . . . . 11 ((ℵ‘𝐴) = ∅ → (𝐵m (ℵ‘𝐴)) = (𝐵m ∅))
99 map0e 8920 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝐵m ∅) = 1o)
10054, 99ax-mp 5 . . . . . . . . . . 11 (𝐵m ∅) = 1o
10198, 100eqtrdi 2790 . . . . . . . . . 10 ((ℵ‘𝐴) = ∅ → (𝐵m (ℵ‘𝐴)) = 1o)
102101fveq2d 6910 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (card‘(𝐵m (ℵ‘𝐴))) = (card‘1o))
103 1onn 8676 . . . . . . . . . 10 1o ∈ ω
104 cardnn 10000 . . . . . . . . . 10 (1o ∈ ω → (card‘1o) = 1o)
105103, 104ax-mp 5 . . . . . . . . 9 (card‘1o) = 1o
106102, 105eqtrdi 2790 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (card‘(𝐵m (ℵ‘𝐴))) = 1o)
107106fveq2d 6910 . . . . . . 7 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) = (cf‘1o))
108 df-1o 8504 . . . . . . . . 9 1o = suc ∅
109108fveq2i 6909 . . . . . . . 8 (cf‘1o) = (cf‘suc ∅)
110 0elon 6439 . . . . . . . . 9 ∅ ∈ On
111 cfsuc 10294 . . . . . . . . 9 (∅ ∈ On → (cf‘suc ∅) = 1o)
112110, 111ax-mp 5 . . . . . . . 8 (cf‘suc ∅) = 1o
113109, 112eqtri 2762 . . . . . . 7 (cf‘1o) = 1o
114107, 113eqtrdi 2790 . . . . . 6 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) = 1o)
11597, 114breq12d 5160 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
11696, 115mpbiri 258 . . . 4 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
117116a1d 25 . . 3 ((ℵ‘𝐴) = ∅ → (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
11892, 117syl 17 . 2 𝐴 ∈ On → (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
11987, 118pm2.61i 182 1 (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  cun 3960  wss 3962  c0 4338  𝒫 cpw 4604   class class class wbr 5147  cmpt 5230   × cxp 5686  dom cdm 5688  ran crn 5689  Oncon0 6385  suc csuc 6387   Fn wfn 6557  cfv 6562  (class class class)co 7430  ωcom 7886  1oc1o 8497  2oc2o 8498  m cmap 8864  cen 8980  cdom 8981  csdm 8982  Fincfn 8983  harchar 9593  cardccrd 9972  cale 9973  cfccf 9974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-ac2 10500
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-smo 8384  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-har 9594  df-card 9976  df-aleph 9977  df-cf 9978  df-acn 9979  df-ac 10153
This theorem is referenced by:  alephom  10622
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