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Theorem cfpwsdom 9852
Description: A corollary of Konig's Theorem konigth 9837. 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 7048 . . . . . . . . 9 (𝐵𝑚 (ℵ‘𝐴)) ∈ V
21cardid 9815 . . . . . . . 8 (card‘(𝐵𝑚 (ℵ‘𝐴))) ≈ (𝐵𝑚 (ℵ‘𝐴))
32ensymi 8407 . . . . . . 7 (𝐵𝑚 (ℵ‘𝐴)) ≈ (card‘(𝐵𝑚 (ℵ‘𝐴)))
4 fvex 6551 . . . . . . . . . . . . . 14 (ℵ‘𝐴) ∈ V
54canth2 8517 . . . . . . . . . . . . 13 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
64pw2en 8471 . . . . . . . . . . . . 13 𝒫 (ℵ‘𝐴) ≈ (2o𝑚 (ℵ‘𝐴))
7 sdomentr 8498 . . . . . . . . . . . . 13 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2o𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2o𝑚 (ℵ‘𝐴)))
85, 6, 7mp2an 688 . . . . . . . . . . . 12 (ℵ‘𝐴) ≺ (2o𝑚 (ℵ‘𝐴))
9 mapdom1 8529 . . . . . . . . . . . 12 (2o𝐵 → (2o𝑚 (ℵ‘𝐴)) ≼ (𝐵𝑚 (ℵ‘𝐴)))
10 sdomdomtr 8497 . . . . . . . . . . . 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 9833 . . . . . . . . . . . . . . . . 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 8962 . . . . . . . . . . . . . . . 16 ((𝐵𝑚 (ℵ‘𝐴)) ∈ Fin → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
1513, 14sylbir 236 . . . . . . . . . . . . . . 15 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (𝐵𝑚 (ℵ‘𝐴)) ≼ ω)
16 alephgeom 9354 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 alephon 9341 . . . . . . . . . . . . . . . . 17 (ℵ‘𝐴) ∈ On
18 ssdomg 8403 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
1917, 18ax-mp 5 . . . . . . . . . . . . . . . 16 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
2016, 19sylbi 218 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21 domtr 8410 . . . . . . . . . . . . . . 15 (((𝐵𝑚 (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
2215, 20, 21syl2an 595 . . . . . . . . . . . . . 14 (((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
23 domnsym 8490 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 9234 . . . . . . . . . . . 12 (card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴)))
28 iscard3 9365 . . . . . . . . . . . . 13 ((card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴))) ↔ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29 elun 4046 . . . . . . . . . . . . 13 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ) ↔ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
30 df-or 843 . . . . . . . . . . . . 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 298 . . . . . . . . . . . 12 ((card‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (card‘(𝐵𝑚 (ℵ‘𝐴))) ↔ (¬ (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ω → (card‘(𝐵𝑚 (ℵ‘𝐴))) ∈ ran ℵ))
3227, 31mpbi 231 . . . . . . . . . . 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 9337 . . . . . . . . . . 11 ℵ Fn On
35 fvelrnb 6594 . . . . . . . . . . 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 252 . . . . . . . . 9 (𝐴 ∈ On → (2o𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴)))))
38 eqid 2795 . . . . . . . . . . . 12 (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧𝑦)))
3938pwcfsdom 9851 . . . . . . . . . . 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 6538 . . . . . . . . . . . . 13 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
4240, 41oveq12d 7034 . . . . . . . . . . . 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 4975 . . . . . . . . . . 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 234 . . . . . . . . . 10 ((ℵ‘𝑥) = (card‘(𝐵𝑚 (ℵ‘𝐴))) → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
4544rexlimivw 3245 . . . . . . . . 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 407 . . . . . . 7 ((𝐴 ∈ On ∧ 2o𝐵) → (card‘(𝐵𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
48 ensdomtr 8500 . . . . . . 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 6551 . . . . . . . . 9 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ∈ V
5150enref 8390 . . . . . . . 8 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))
52 mapen 8528 . . . . . . . 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 688 . . . . . . 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 8530 . . . . . . . 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 1453 . . . . . . 7 ((𝐵𝑚 (ℵ‘𝐴)) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
5753, 56entri 8411 . . . . . 6 ((card‘(𝐵𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≈ (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
58 sdomentr 8498 . . . . . 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 8459 . . . . . . . . . 10 ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6116biimpi 217 . . . . . . . . . . 11 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62 infxpen 9286 . . . . . . . . . . 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 8416 . . . . . . . . . 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 595 . . . . . . . . 9 (((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
66 nsuceq0 6146 . . . . . . . . . . 11 suc 1o ≠ ∅
67 dom0 8492 . . . . . . . . . . 11 (suc 1o ≼ ∅ ↔ suc 1o = ∅)
6866, 67nemtbir 3080 . . . . . . . . . 10 ¬ suc 1o ≼ ∅
69 df-2o 7954 . . . . . . . . . . . . . 14 2o = suc 1o
7069breq1i 4969 . . . . . . . . . . . . 13 (2o𝐵 ↔ suc 1o𝐵)
71 breq2 4966 . . . . . . . . . . . . 13 (𝐵 = ∅ → (suc 1o𝐵 ↔ suc 1o ≼ ∅))
7270, 71syl5bb 284 . . . . . . . . . . . 12 (𝐵 = ∅ → (2o𝐵 ↔ suc 1o ≼ ∅))
7372biimpcd 250 . . . . . . . . . . 11 (2o𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
7473adantld 491 . . . . . . . . . 10 (2o𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
7568, 74mtoi 200 . . . . . . . . 9 (2o𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76 mapdom2 8535 . . . . . . . . 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 595 . . . . . . . 8 ((((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2o𝐵) → (𝐵𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))) ≼ (𝐵𝑚 (ℵ‘𝐴)))
78 domnsym 8490 . . . . . . . 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 458 . . . . . 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 9824 . . . . . 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 688 . . . . 5 ((cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
8584biimpri 229 . . . 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 413 . 2 (𝐴 ∈ On → (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴))))))
88 fndm 6325 . . . . . 6 (ℵ Fn On → dom ℵ = On)
8934, 88ax-mp 5 . . . . 5 dom ℵ = On
9089eleq2i 2874 . . . 4 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91 ndmfv 6568 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
9290, 91sylnbir 332 . . 3 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93 1n0 7970 . . . . . 6 1o ≠ ∅
94 1oex 7961 . . . . . . 7 1o ∈ V
95940sdom 8495 . . . . . 6 (∅ ≺ 1o ↔ 1o ≠ ∅)
9693, 95mpbir 232 . . . . 5 ∅ ≺ 1o
97 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98 oveq2 7024 . . . . . . . . . . 11 ((ℵ‘𝐴) = ∅ → (𝐵𝑚 (ℵ‘𝐴)) = (𝐵𝑚 ∅))
99 map0e 8296 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝐵𝑚 ∅) = 1o)
10054, 99ax-mp 5 . . . . . . . . . . 11 (𝐵𝑚 ∅) = 1o
10198, 100syl6eq 2847 . . . . . . . . . 10 ((ℵ‘𝐴) = ∅ → (𝐵𝑚 (ℵ‘𝐴)) = 1o)
102101fveq2d 6542 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (card‘(𝐵𝑚 (ℵ‘𝐴))) = (card‘1o))
103 1onn 8115 . . . . . . . . . 10 1o ∈ ω
104 cardnn 9238 . . . . . . . . . 10 (1o ∈ ω → (card‘1o) = 1o)
105103, 104ax-mp 5 . . . . . . . . 9 (card‘1o) = 1o
106102, 105syl6eq 2847 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (card‘(𝐵𝑚 (ℵ‘𝐴))) = 1o)
107106fveq2d 6542 . . . . . . 7 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = (cf‘1o))
108 df-1o 7953 . . . . . . . . 9 1o = suc ∅
109108fveq2i 6541 . . . . . . . 8 (cf‘1o) = (cf‘suc ∅)
110 0elon 6119 . . . . . . . . 9 ∅ ∈ On
111 cfsuc 9525 . . . . . . . . 9 (∅ ∈ On → (cf‘suc ∅) = 1o)
112110, 111ax-mp 5 . . . . . . . 8 (cf‘suc ∅) = 1o
113109, 112eqtri 2819 . . . . . . 7 (cf‘1o) = 1o
114107, 113syl6eq 2847 . . . . . 6 ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) = 1o)
11597, 114breq12d 4975 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
11696, 115mpbiri 259 . . . 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 183 1 (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842   = wceq 1522  wcel 2081  wne 2984  wrex 3106  Vcvv 3437  cun 3857  wss 3859  c0 4211  𝒫 cpw 4453   class class class wbr 4962  cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444  Oncon0 6066  suc csuc 6068   Fn wfn 6220  cfv 6225  (class class class)co 7016  ωcom 7436  1oc1o 7946  2oc2o 7947  𝑚 cmap 8256  cen 8354  cdom 8355  csdm 8356  Fincfn 8357  harchar 8866  cardccrd 9210  cale 9211  cfccf 9212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-ac2 9731
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-smo 7835  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-oi 8820  df-har 8868  df-card 9214  df-aleph 9215  df-cf 9216  df-acn 9217  df-ac 9388
This theorem is referenced by:  alephom  9853
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