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Theorem alephval2 9432
Description: An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.)
Assertion
Ref Expression
alephval2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem alephval2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 alephordi 8935 . . . . . 6 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
21ralrimiv 2994 . . . . 5 (𝐴 ∈ On → ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3 alephon 8930 . . . . 5 (ℵ‘𝐴) ∈ On
42, 3jctil 559 . . . 4 (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5 breq2 4689 . . . . . 6 (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
65ralbidv 3015 . . . . 5 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
76elrab 3396 . . . 4 ((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
84, 7sylibr 224 . . 3 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
98adantr 480 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
10 cardsdomelir 8837 . . . . 5 (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
11 alephcard 8931 . . . . . 6 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
1211eqcomi 2660 . . . . 5 (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
1310, 12eleq2s 2748 . . . 4 (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
14 omex 8578 . . . . . 6 ω ∈ V
15 vex 3234 . . . . . 6 𝑧 ∈ V
16 entri3 9419 . . . . . 6 ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧𝑧 ≼ ω))
1714, 15, 16mp2an 708 . . . . 5 (ω ≼ 𝑧𝑧 ≼ ω)
18 carddom 9414 . . . . . . . . . 10 ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
1914, 15, 18mp2an 708 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
20 cardom 8850 . . . . . . . . . 10 (card‘ω) = ω
2120sseq1i 3662 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
2219, 21bitr3i 266 . . . . . . . 8 (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
23 cardidm 8823 . . . . . . . . . 10 (card‘(card‘𝑧)) = (card‘𝑧)
24 cardalephex 8951 . . . . . . . . . 10 (ω ⊆ (card‘𝑧) → ((card‘(card‘𝑧)) = (card‘𝑧) ↔ ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥)))
2523, 24mpbii 223 . . . . . . . . 9 (ω ⊆ (card‘𝑧) → ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥))
26 alephord 8936 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2726ancoms 468 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2815cardid 9407 . . . . . . . . . . . . . . 15 (card‘𝑧) ≈ 𝑧
29 sdomen1 8145 . . . . . . . . . . . . . . 15 ((card‘𝑧) ≈ 𝑧 → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
3028, 29ax-mp 5 . . . . . . . . . . . . . 14 ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴))
31 breq1 4688 . . . . . . . . . . . . . 14 ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
3230, 31syl5rbbr 275 . . . . . . . . . . . . 13 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
3327, 32sylan9bb 736 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴𝑧 ≺ (ℵ‘𝐴)))
34 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
3534breq1d 4695 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
3635rspcv 3336 . . . . . . . . . . . . . . 15 (𝑥𝐴 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
37 sdomirr 8138 . . . . . . . . . . . . . . . 16 ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
38 sdomen2 8146 . . . . . . . . . . . . . . . . . 18 ((card‘𝑧) ≈ 𝑧 → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧))
3928, 38ax-mp 5 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧)
40 breq2 4689 . . . . . . . . . . . . . . . . 17 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4139, 40syl5bbr 274 . . . . . . . . . . . . . . . 16 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4237, 41mtbiri 316 . . . . . . . . . . . . . . 15 ((card‘𝑧) = (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ 𝑧)
4336, 42nsyli 155 . . . . . . . . . . . . . 14 (𝑥𝐴 → ((card‘𝑧) = (ℵ‘𝑥) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4443com12 32 . . . . . . . . . . . . 13 ((card‘𝑧) = (ℵ‘𝑥) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4544adantl 481 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4633, 45sylbird 250 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4746exp31 629 . . . . . . . . . 10 (𝐴 ∈ On → (𝑥 ∈ On → ((card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))))
4847rexlimdv 3059 . . . . . . . . 9 (𝐴 ∈ On → (∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
4925, 48syl5 34 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ (card‘𝑧) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
5022, 49syl5bi 232 . . . . . . 7 (𝐴 ∈ On → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
5150adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
52 ne0i 3954 . . . . . . . . . . . 12 (∅ ∈ 𝐴𝐴 ≠ ∅)
53 onelon 5786 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
54 alephgeom 8943 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
55 alephon 8930 . . . . . . . . . . . . . . . . . . 19 (ℵ‘𝑦) ∈ On
56 ssdomg 8043 . . . . . . . . . . . . . . . . . . 19 ((ℵ‘𝑦) ∈ On → (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦)))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . 18 (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦))
5854, 57sylbi 207 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ On → ω ≼ (ℵ‘𝑦))
59 domtr 8050 . . . . . . . . . . . . . . . . 17 ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
6058, 59sylan2 490 . . . . . . . . . . . . . . . 16 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
61 domnsym 8127 . . . . . . . . . . . . . . . 16 (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6260, 61syl 17 . . . . . . . . . . . . . . 15 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6353, 62sylan2 490 . . . . . . . . . . . . . 14 ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6463expr 642 . . . . . . . . . . . . 13 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
6564ralrimiv 2994 . . . . . . . . . . . 12 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
66 r19.2z 4093 . . . . . . . . . . . . 13 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
6766ex 449 . . . . . . . . . . . 12 (𝐴 ≠ ∅ → (∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
6852, 65, 67syl2im 40 . . . . . . . . . . 11 (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
69 rexnal 3024 . . . . . . . . . . 11 (∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 ↔ ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
7068, 69syl6ib 241 . . . . . . . . . 10 (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7170com12 32 . . . . . . . . 9 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7271expimpd 628 . . . . . . . 8 (𝑧 ≼ ω → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7372a1d 25 . . . . . . 7 (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7473com3r 87 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7551, 74jaod 394 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((ω ≼ 𝑧𝑧 ≼ ω) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7617, 75mpi 20 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
77 breq2 4689 . . . . . . . 8 (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
7877ralbidv 3015 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7978elrab 3396 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ (𝑧 ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
8079simprbi 479 . . . . 5 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} → ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
8180con3i 150 . . . 4 (¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
8213, 76, 81syl56 36 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}))
8382ralrimiv 2994 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
84 ssrab2 3720 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
85 oneqmini 5814 . . 3 ({𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On → (((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}))
8684, 85ax-mp 5 . 2 (((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
879, 83, 86syl2anc 694 1 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948   cint 4507   class class class wbr 4685  Oncon0 5761  cfv 5926  ωcom 7107  cen 7994  cdom 7995  csdm 7996  cardccrd 8799  cale 8800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-card 8803  df-aleph 8804  df-ac 8977
This theorem is referenced by: (None)
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