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Theorem minregex 43436
Description: Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of which is greater to or equal to 𝐴. This proof uses AC. (Contributed by RP, 23-Nov-2023.)
Assertion
Ref Expression
minregex (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem minregex
StepHypRef Expression
1 eldif 3980 . . . . . . 7 (𝐴 ∈ (ran card ∖ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
2 omelon 9711 . . . . . . . . . 10 ω ∈ On
3 cardon 10009 . . . . . . . . . . 11 (card‘𝐴) ∈ On
4 eleq1 2826 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
53, 4mpbii 233 . . . . . . . . . 10 ((card‘𝐴) = 𝐴𝐴 ∈ On)
6 ontri1 6428 . . . . . . . . . 10 ((ω ∈ On ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
72, 5, 6sylancr 586 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
87pm5.32i 574 . . . . . . . 8 (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) ↔ ((card‘𝐴) = 𝐴 ∧ ¬ 𝐴 ∈ ω))
9 iscard4 43435 . . . . . . . . 9 ((card‘𝐴) = 𝐴𝐴 ∈ ran card)
109anbi1i 623 . . . . . . . 8 (((card‘𝐴) = 𝐴 ∧ ¬ 𝐴 ∈ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
118, 10bitr2i 276 . . . . . . 7 ((𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω) ↔ ((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴))
12 ancom 460 . . . . . . 7 (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
131, 11, 123bitri 297 . . . . . 6 (𝐴 ∈ (ran card ∖ ω) ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
1413biimpi 216 . . . . 5 (𝐴 ∈ (ran card ∖ ω) → (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
15 cardalephex 10155 . . . . . . . 8 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
1615biimpa 476 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
17 eqimss 4061 . . . . . . . . 9 (𝐴 = (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘𝑥))
1817a1i 11 . . . . . . . 8 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (𝐴 = (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘𝑥)))
1918reximdv 3172 . . . . . . 7 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥)))
2016, 19mpd 15 . . . . . 6 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
21 onintrab2 7829 . . . . . 6 (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
2220, 21sylib 218 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
23 simpr 484 . . . . . . 7 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
24 onsuc 7843 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
2523, 24syl 17 . . . . . 6 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
26 eloni 6404 . . . . . . . . 9 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
2723, 26syl 17 . . . . . . . 8 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
28 0elsuc 7867 . . . . . . . 8 (Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
2927, 28syl 17 . . . . . . 7 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → ∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
30 cardaleph 10154 . . . . . . . . 9 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
3130adantr 480 . . . . . . . 8 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
32 sssucid 6474 . . . . . . . . 9 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
33 alephord3 10143 . . . . . . . . . 10 (( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
3423, 24, 33syl2anc2 584 . . . . . . . . 9 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
3532, 34mpbii 233 . . . . . . . 8 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
3631, 35eqsstrd 4041 . . . . . . 7 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
37 alephreg 10647 . . . . . . . 8 (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
3837a1i 11 . . . . . . 7 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
3929, 36, 383jca 1128 . . . . . 6 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
4025, 39jca 511 . . . . 5 (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
4114, 22, 40syl2anc2 584 . . . 4 (𝐴 ∈ (ran card ∖ ω) → (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
4214, 16syl 17 . . . . . . . 8 (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
4317a1i 11 . . . . . . . . 9 (𝐴 ∈ (ran card ∖ ω) → (𝐴 = (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘𝑥)))
4443reximdv 3172 . . . . . . . 8 (𝐴 ∈ (ran card ∖ ω) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥)))
4542, 44mpd 15 . . . . . . 7 (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
4645, 21sylib 218 . . . . . 6 (𝐴 ∈ (ran card ∖ ω) → {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
4746, 24syl 17 . . . . 5 (𝐴 ∈ (ran card ∖ ω) → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
48 sbcan 3851 . . . . . 6 ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ∧ [suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
49 sbcel1v 3869 . . . . . . . 8 ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ↔ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
5049a1i 11 . . . . . . 7 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ↔ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On))
51 sbc3an 3868 . . . . . . . 8 ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦[suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ∧ [suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
52 sbcel2gv 3870 . . . . . . . . 9 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
53 sbcssg 4543 . . . . . . . . . 10 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ↔ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦𝐴suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦)))
54 csbconstg 3934 . . . . . . . . . . 11 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦𝐴 = 𝐴)
55 csbfv2g 6968 . . . . . . . . . . . 12 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦𝑦))
56 csbvarg 4453 . . . . . . . . . . . . 13 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦𝑦 = suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
5756fveq2d 6923 . . . . . . . . . . . 12 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦𝑦) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
5855, 57eqtrd 2774 . . . . . . . . . . 11 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
5954, 58sseq12d 4036 . . . . . . . . . 10 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦𝐴suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
6053, 59bitrd 279 . . . . . . . . 9 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
61 sbceqg 4431 . . . . . . . . . 10 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦) ↔ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(cf‘(ℵ‘𝑦)) = suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦)))
62 csbfv2g 6968 . . . . . . . . . . . 12 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(cf‘(ℵ‘𝑦)) = (cf‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦)))
6358fveq2d 6923 . . . . . . . . . . . 12 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (cf‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦)) = (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
6462, 63eqtrd 2774 . . . . . . . . . . 11 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(cf‘(ℵ‘𝑦)) = (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
6564, 58eqeq12d 2750 . . . . . . . . . 10 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(cf‘(ℵ‘𝑦)) = suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦(ℵ‘𝑦) ↔ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
6661, 65bitrd 279 . . . . . . . . 9 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦) ↔ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
6752, 60, 663anbi123d 1436 . . . . . . . 8 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦[suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ∧ [suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
6851, 67bitrid 283 . . . . . . 7 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
6950, 68anbi12d 631 . . . . . 6 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ∧ [suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))))
7048, 69bitrid 283 . . . . 5 (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))))
7147, 70syl 17 . . . 4 (𝐴 ∈ (ran card ∖ ω) → ([suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ (suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))))
7241, 71mpbird 257 . . 3 (𝐴 ∈ (ran card ∖ ω) → [suc {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
7372spesbcd 3899 . 2 (𝐴 ∈ (ran card ∖ ω) → ∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
74 onintrab2 7829 . . 3 (∃𝑦 ∈ On (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ∈ On)
75 df-rex 3073 . . 3 (∃𝑦 ∈ On (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ ∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
76 risset 3234 . . 3 ( {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ∈ On ↔ ∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
7774, 75, 763bitr3i 301 . 2 (∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ ∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
7873, 77sylib 218 1 (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wex 1777  wcel 2103  wrex 3072  {crab 3438  [wsbc 3798  csb 3915  cdif 3967  wss 3970  c0 4347   cint 4972  ran crn 5700  Ord word 6393  Oncon0 6394  suc csuc 6396  cfv 6572  ωcom 7899  cardccrd 10000  cale 10001  cfccf 10002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-inf2 9706  ax-ac2 10528
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-isom 6581  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-er 8759  df-map 8882  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-oi 9575  df-har 9622  df-card 10004  df-aleph 10005  df-cf 10006  df-acn 10007  df-ac 10181
This theorem is referenced by:  minregex2  43437
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