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Theorem cfpwsdom 9995
 Description: A corollary of Konig's Theorem konigth 9980. 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 7168 . . . . . . . . 9 (𝐵m (ℵ‘𝐴)) ∈ V
21cardid 9958 . . . . . . . 8 (card‘(𝐵m (ℵ‘𝐴))) ≈ (𝐵m (ℵ‘𝐴))
32ensymi 8542 . . . . . . 7 (𝐵m (ℵ‘𝐴)) ≈ (card‘(𝐵m (ℵ‘𝐴)))
4 fvex 6658 . . . . . . . . . . . . . 14 (ℵ‘𝐴) ∈ V
54canth2 8654 . . . . . . . . . . . . 13 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
64pw2en 8607 . . . . . . . . . . . . 13 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))
7 sdomentr 8635 . . . . . . . . . . . . 13 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)))
85, 6, 7mp2an 691 . . . . . . . . . . . 12 (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴))
9 mapdom1 8666 . . . . . . . . . . . 12 (2o𝐵 → (2om (ℵ‘𝐴)) ≼ (𝐵m (ℵ‘𝐴)))
10 sdomdomtr 8634 . . . . . . . . . . . 12 (((ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)) ∧ (2om (ℵ‘𝐴)) ≼ (𝐵m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
118, 9, 10sylancr 590 . . . . . . . . . . 11 (2o𝐵 → (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)))
12 ficard 9976 . . . . . . . . . . . . . . . . 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 9100 . . . . . . . . . . . . . . . 16 ((𝐵m (ℵ‘𝐴)) ∈ Fin → (𝐵m (ℵ‘𝐴)) ≼ ω)
1513, 14sylbir 238 . . . . . . . . . . . . . . 15 ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (𝐵m (ℵ‘𝐴)) ≼ ω)
16 alephgeom 9493 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 alephon 9480 . . . . . . . . . . . . . . . . 17 (ℵ‘𝐴) ∈ On
18 ssdomg 8538 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
1917, 18ax-mp 5 . . . . . . . . . . . . . . . 16 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
2016, 19sylbi 220 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21 domtr 8545 . . . . . . . . . . . . . . 15 (((𝐵m (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
2215, 20, 21syl2an 598 . . . . . . . . . . . . . 14 (((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
23 domnsym 8627 . . . . . . . . . . . . . 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 417 . . . . . . . . . . . 12 (𝐴 ∈ On → ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω → ¬ (ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴))))
2625con2d 136 . . . . . . . . . . 11 (𝐴 ∈ On → ((ℵ‘𝐴) ≺ (𝐵m (ℵ‘𝐴)) → ¬ (card‘(𝐵m (ℵ‘𝐴))) ∈ ω))
27 cardidm 9372 . . . . . . . . . . . 12 (card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴)))
28 iscard3 9504 . . . . . . . . . . . . 13 ((card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴))) ↔ (card‘(𝐵m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29 elun 4076 . . . . . . . . . . . . 13 ((card‘(𝐵m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ) ↔ ((card‘(𝐵m (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
30 df-or 845 . . . . . . . . . . . . 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 300 . . . . . . . . . . . 12 ((card‘(card‘(𝐵m (ℵ‘𝐴)))) = (card‘(𝐵m (ℵ‘𝐴))) ↔ (¬ (card‘(𝐵m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵m (ℵ‘𝐴))) ∈ ran ℵ))
3227, 31mpbi 233 . . . . . . . . . . 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 9476 . . . . . . . . . . 11 ℵ Fn On
35 fvelrnb 6701 . . . . . . . . . . 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, 36syl6ib 254 . . . . . . . . 9 (𝐴 ∈ On → (2o𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴)))))
38 eqid 2798 . . . . . . . . . . . 12 (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦)))
3938pwcfsdom 9994 . . . . . . . . . . 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 6645 . . . . . . . . . . . . 13 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
4240, 41oveq12d 7153 . . . . . . . . . . . 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 5043 . . . . . . . . . . 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 236 . . . . . . . . . 10 ((ℵ‘𝑥) = (card‘(𝐵m (ℵ‘𝐴))) → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
4544rexlimivw 3241 . . . . . . . . 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 410 . . . . . . 7 ((𝐴 ∈ On ∧ 2o𝐵) → (card‘(𝐵m (ℵ‘𝐴))) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
48 ensdomtr 8637 . . . . . . 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 590 . . . . . 6 ((𝐴 ∈ On ∧ 2o𝐵) → (𝐵m (ℵ‘𝐴)) ≺ ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
50 fvex 6658 . . . . . . . . 9 (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ∈ V
5150enref 8525 . . . . . . . 8 (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵m (ℵ‘𝐴))))
52 mapen 8665 . . . . . . . 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 691 . . . . . . 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 8667 . . . . . . . 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 1458 . . . . . . 7 ((𝐵m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≈ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
5753, 56entri 8546 . . . . . 6 ((card‘(𝐵m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≈ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
58 sdomentr 8635 . . . . . 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 589 . . . . 5 ((𝐴 ∈ On ∧ 2o𝐵) → (𝐵m (ℵ‘𝐴)) ≺ (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))))
604xpdom2 8595 . . . . . . . . . 10 ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6116biimpi 219 . . . . . . . . . . 11 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62 infxpen 9425 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6317, 61, 62sylancr 590 . . . . . . . . . 10 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
64 domentr 8551 . . . . . . . . . 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 598 . . . . . . . . 9 (((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
66 nsuceq0 6239 . . . . . . . . . . 11 suc 1o ≠ ∅
67 dom0 8629 . . . . . . . . . . 11 (suc 1o ≼ ∅ ↔ suc 1o = ∅)
6866, 67nemtbir 3082 . . . . . . . . . 10 ¬ suc 1o ≼ ∅
69 df-2o 8086 . . . . . . . . . . . . . 14 2o = suc 1o
7069breq1i 5037 . . . . . . . . . . . . 13 (2o𝐵 ↔ suc 1o𝐵)
71 breq2 5034 . . . . . . . . . . . . 13 (𝐵 = ∅ → (suc 1o𝐵 ↔ suc 1o ≼ ∅))
7270, 71syl5bb 286 . . . . . . . . . . . 12 (𝐵 = ∅ → (2o𝐵 ↔ suc 1o ≼ ∅))
7372biimpcd 252 . . . . . . . . . . 11 (2o𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
7473adantld 494 . . . . . . . . . 10 (2o𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
7568, 74mtoi 202 . . . . . . . . 9 (2o𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76 mapdom2 8672 . . . . . . . . 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 598 . . . . . . . 8 ((((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2o𝐵) → (𝐵m ((ℵ‘𝐴) × (cf‘(card‘(𝐵m (ℵ‘𝐴)))))) ≼ (𝐵m (ℵ‘𝐴)))
78 domnsym 8627 . . . . . . . 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 461 . . . . . 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 138 . . . 4 ((𝐴 ∈ On ∧ 2o𝐵) → ¬ (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
83 domtri 9967 . . . . . 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 691 . . . . 5 ((cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
8584biimpri 231 . . . 4 (¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))) → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
8682, 85nsyl2 143 . . 3 ((𝐴 ∈ On ∧ 2o𝐵) → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
8786ex 416 . 2 (𝐴 ∈ On → (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴))))))
88 fndm 6425 . . . . . 6 (ℵ Fn On → dom ℵ = On)
8934, 88ax-mp 5 . . . . 5 dom ℵ = On
9089eleq2i 2881 . . . 4 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91 ndmfv 6675 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
9290, 91sylnbir 334 . . 3 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93 1n0 8102 . . . . . 6 1o ≠ ∅
94 1oex 8093 . . . . . . 7 1o ∈ V
95940sdom 8632 . . . . . 6 (∅ ≺ 1o ↔ 1o ≠ ∅)
9693, 95mpbir 234 . . . . 5 ∅ ≺ 1o
97 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98 oveq2 7143 . . . . . . . . . . 11 ((ℵ‘𝐴) = ∅ → (𝐵m (ℵ‘𝐴)) = (𝐵m ∅))
99 map0e 8429 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝐵m ∅) = 1o)
10054, 99ax-mp 5 . . . . . . . . . . 11 (𝐵m ∅) = 1o
10198, 100eqtrdi 2849 . . . . . . . . . 10 ((ℵ‘𝐴) = ∅ → (𝐵m (ℵ‘𝐴)) = 1o)
102101fveq2d 6649 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (card‘(𝐵m (ℵ‘𝐴))) = (card‘1o))
103 1onn 8248 . . . . . . . . . 10 1o ∈ ω
104 cardnn 9376 . . . . . . . . . 10 (1o ∈ ω → (card‘1o) = 1o)
105103, 104ax-mp 5 . . . . . . . . 9 (card‘1o) = 1o
106102, 105eqtrdi 2849 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (card‘(𝐵m (ℵ‘𝐴))) = 1o)
107106fveq2d 6649 . . . . . . 7 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) = (cf‘1o))
108 df-1o 8085 . . . . . . . . 9 1o = suc ∅
109108fveq2i 6648 . . . . . . . 8 (cf‘1o) = (cf‘suc ∅)
110 0elon 6212 . . . . . . . . 9 ∅ ∈ On
111 cfsuc 9668 . . . . . . . . 9 (∅ ∈ On → (cf‘suc ∅) = 1o)
112110, 111ax-mp 5 . . . . . . . 8 (cf‘suc ∅) = 1o
113109, 112eqtri 2821 . . . . . . 7 (cf‘1o) = 1o
114107, 113eqtrdi 2849 . . . . . 6 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵m (ℵ‘𝐴)))) = 1o)
11597, 114breq12d 5043 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
11696, 115mpbiri 261 . . . 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 185 1 (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517  dom cdm 5519  ran crn 5520  Oncon0 6159  suc csuc 6161   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135  ωcom 7560  1oc1o 8078  2oc2o 8079   ↑m cmap 8389   ≈ cen 8489   ≼ cdom 8490   ≺ csdm 8491  Fincfn 8492  harchar 9004  cardccrd 9348  ℵcale 9349  cfccf 9350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-ac2 9874 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-smo 7966  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-har 9005  df-card 9352  df-aleph 9353  df-cf 9354  df-acn 9355  df-ac 9527 This theorem is referenced by:  alephom  9996
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