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Theorem cfpwsdom 10623
Description: A corollary of Konig's Theorem konigth 10608. 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 7456 . . . . . . . . 9 (𝐵m (ℵ‘𝐴)) ∈ V
21cardid 10586 . . . . . . . 8 (card‘(𝐵m (ℵ‘𝐴))) ≈ (𝐵m (ℵ‘𝐴))
32ensymi 9034 . . . . . . 7 (𝐵m (ℵ‘𝐴)) ≈ (card‘(𝐵m (ℵ‘𝐴)))
4 fvex 6913 . . . . . . . . . . . . . 14 (ℵ‘𝐴) ∈ V
54canth2 9167 . . . . . . . . . . . . 13 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
64pw2en 9116 . . . . . . . . . . . . 13 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))
7 sdomentr 9148 . . . . . . . . . . . . 13 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)))
85, 6, 7mp2an 690 . . . . . . . . . . . 12 (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴))
9 mapdom1 9179 . . . . . . . . . . . 12 (2o𝐵 → (2om (ℵ‘𝐴)) ≼ (𝐵m (ℵ‘𝐴)))
10 sdomdomtr 9147 . . . . . . . . . . . 12 (((ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)) ∧ (2om (ℵ‘𝐴)) ≼ (𝐵m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
118, 9, 10sylancr 585 . . . . . . . . . . 11 (2o𝐵 → (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
12 ficard 10604 . . . . . . . . . . . . . . . . 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 9692 . . . . . . . . . . . . . . . 16 ((𝐵m (ℵ‘𝐴)) ∈ Fin → (𝐵m (ℵ‘𝐴)) ≼ ω)
1513, 14sylbir 234 . . . . . . . . . . . . . . 15 ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (𝐵m (ℵ‘𝐴)) ≼ ω)
16 alephgeom 10121 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 alephon 10108 . . . . . . . . . . . . . . . . 17 (ℵ‘𝐴) ∈ On
18 ssdomg 9030 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
1917, 18ax-mp 5 . . . . . . . . . . . . . . . 16 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
2016, 19sylbi 216 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21 domtr 9037 . . . . . . . . . . . . . . 15 (((𝐵m (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
2215, 20, 21syl2an 594 . . . . . . . . . . . . . 14 (((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
23 domnsym 9136 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 9998 . . . . . . . . . . . 12 (card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴)))
28 iscard3 10132 . . . . . . . . . . . . 13 ((card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴))) ↔ (card‘(𝐵m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29 elun 4147 . . . . . . . . . . . . 13 ((card‘(𝐵m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ) ↔ ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
30 df-or 846 . . . . . . . . . . . . 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 296 . . . . . . . . . . . 12 ((card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴))) ↔ (¬ (card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
3227, 31mpbi 229 . . . . . . . . . . 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 10104 . . . . . . . . . . 11 ℵ Fn On
35 fvelrnb 6962 . . . . . . . . . . 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 250 . . . . . . . . 9 (𝐴 ∈ On → (2o𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴)))))
38 eqid 2725 . . . . . . . . . . . 12 (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦)))
3938pwcfsdom 10622 . . . . . . . . . . 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 6900 . . . . . . . . . . . . 13 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
4240, 41oveq12d 7441 . . . . . . . . . . . 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 5165 . . . . . . . . . . 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 232 . . . . . . . . . 10 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
4544rexlimivw 3140 . . . . . . . . 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 405 . . . . . . 7 ((𝐴 ∈ On ∧ 2o𝐵) → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
48 ensdomtr 9150 . . . . . . 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 585 . . . . . 6 ((𝐴 ∈ On ∧ 2o𝐵) → (𝐵m (ℵ‘𝐴)) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
50 fvex 6913 . . . . . . . . 9 (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ∈ V
5150enref 9015 . . . . . . . 8 (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵m (ℵ‘𝐴))))
52 mapen 9178 . . . . . . . 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 690 . . . . . . 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 9180 . . . . . . . 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 1457 . . . . . . 7 ((𝐵m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≈ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
5753, 56entri 9038 . . . . . 6 ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≈ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
58 sdomentr 9148 . . . . . 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 584 . . . . 5 ((𝐴 ∈ On ∧ 2o𝐵) → (𝐵m (ℵ‘𝐴)) ≺ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))))
604xpdom2 9104 . . . . . . . . . 10 ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6116biimpi 215 . . . . . . . . . . 11 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62 infxpen 10053 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6317, 61, 62sylancr 585 . . . . . . . . . 10 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
64 domentr 9043 . . . . . . . . . 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 594 . . . . . . . . 9 (((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
66 nsuceq0 6458 . . . . . . . . . . 11 suc 1o ≠ ∅
67 dom0 9139 . . . . . . . . . . 11 (suc 1o ≼ ∅ ↔ suc 1o = ∅)
6866, 67nemtbir 3027 . . . . . . . . . 10 ¬ suc 1o ≼ ∅
69 df-2o 8496 . . . . . . . . . . . . . 14 2o = suc 1o
7069breq1i 5159 . . . . . . . . . . . . 13 (2o𝐵 ↔ suc 1o𝐵)
71 breq2 5156 . . . . . . . . . . . . 13 (𝐵 = ∅ → (suc 1o𝐵 ↔ suc 1o ≼ ∅))
7270, 71bitrid 282 . . . . . . . . . . . 12 (𝐵 = ∅ → (2o𝐵 ↔ suc 1o ≼ ∅))
7372biimpcd 248 . . . . . . . . . . 11 (2o𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
7473adantld 489 . . . . . . . . . 10 (2o𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
7568, 74mtoi 198 . . . . . . . . 9 (2o𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76 mapdom2 9185 . . . . . . . . 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 594 . . . . . . . 8 ((((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2o𝐵) → (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))) ≼ (𝐵m (ℵ‘𝐴)))
78 domnsym 9136 . . . . . . . 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 456 . . . . . 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 10595 . . . . . 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 690 . . . . 5 ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
8584biimpri 227 . . . 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 411 . 2 (𝐴 ∈ On → (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
88 fndm 6662 . . . . . 6 (ℵ Fn On → dom ℵ = On)
8934, 88ax-mp 5 . . . . 5 dom ℵ = On
9089eleq2i 2817 . . . 4 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91 ndmfv 6935 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
9290, 91sylnbir 330 . . 3 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93 1n0 8517 . . . . . 6 1o ≠ ∅
94 1oex 8505 . . . . . . 7 1o ∈ V
95940sdom 9144 . . . . . 6 (∅ ≺ 1o ↔ 1o ≠ ∅)
9693, 95mpbir 230 . . . . 5 ∅ ≺ 1o
97 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98 oveq2 7431 . . . . . . . . . . 11 ((ℵ‘𝐴) = ∅ → (𝐵m (ℵ‘𝐴)) = (𝐵m ∅))
99 map0e 8910 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝐵m ∅) = 1o)
10054, 99ax-mp 5 . . . . . . . . . . 11 (𝐵m ∅) = 1o
10198, 100eqtrdi 2781 . . . . . . . . . 10 ((ℵ‘𝐴) = ∅ → (𝐵m (ℵ‘𝐴)) = 1o)
102101fveq2d 6904 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (card‘(𝐵m (ℵ‘𝐴))) = (card‘1o))
103 1onn 8669 . . . . . . . . . 10 1o ∈ ω
104 cardnn 10002 . . . . . . . . . 10 (1o ∈ ω → (card‘1o) = 1o)
105103, 104ax-mp 5 . . . . . . . . 9 (card‘1o) = 1o
106102, 105eqtrdi 2781 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (card‘(𝐵m (ℵ‘𝐴))) = 1o)
107106fveq2d 6904 . . . . . . 7 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) = (cf‘1o))
108 df-1o 8495 . . . . . . . . 9 1o = suc ∅
109108fveq2i 6903 . . . . . . . 8 (cf‘1o) = (cf‘suc ∅)
110 0elon 6429 . . . . . . . . 9 ∅ ∈ On
111 cfsuc 10296 . . . . . . . . 9 (∅ ∈ On → (cf‘suc ∅) = 1o)
112110, 111ax-mp 5 . . . . . . . 8 (cf‘suc ∅) = 1o
113109, 112eqtri 2753 . . . . . . 7 (cf‘1o) = 1o
114107, 113eqtrdi 2781 . . . . . 6 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) = 1o)
11597, 114breq12d 5165 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
11696, 115mpbiri 257 . . . 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 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2929  wrex 3059  Vcvv 3461  cun 3944  wss 3946  c0 4324  𝒫 cpw 4606   class class class wbr 5152  cmpt 5235   × cxp 5679  dom cdm 5681  ran crn 5682  Oncon0 6375  suc csuc 6377   Fn wfn 6548  cfv 6553  (class class class)co 7423  ωcom 7875  1oc1o 8488  2oc2o 8489  m cmap 8854  cen 8970  cdom 8971  csdm 8972  Fincfn 8973  harchar 9595  cardccrd 9974  cale 9975  cfccf 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-inf2 9680  ax-ac2 10502
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-se 5637  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-1st 8002  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-smo 8375  df-recs 8400  df-rdg 8439  df-1o 8495  df-2o 8496  df-er 8733  df-map 8856  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-oi 9549  df-har 9596  df-card 9978  df-aleph 9979  df-cf 9980  df-acn 9981  df-ac 10155
This theorem is referenced by:  alephom  10624
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