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Theorem oeeulem 7890
Description: Lemma for oeeu 7892. (Contributed by Mario Carneiro, 28-Feb-2013.)
Hypothesis
Ref Expression
oeeu.1 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}
Assertion
Ref Expression
oeeulem ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝑋 ∈ On ∧ (𝐴𝑜 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝑋)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem oeeulem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oeeu.1 . . 3 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}
2 eldifi 3896 . . . . . . . 8 (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ∈ On)
32adantl 473 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ On)
4 suceloni 7215 . . . . . . 7 (𝐵 ∈ On → suc 𝐵 ∈ On)
53, 4syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ∈ On)
6 oeworde 7882 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴𝑜 suc 𝐵))
75, 6syldan 585 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ⊆ (𝐴𝑜 suc 𝐵))
8 sucidg 5988 . . . . . . . 8 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
93, 8syl 17 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ suc 𝐵)
107, 9sseldd 3764 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴𝑜 suc 𝐵))
11 oveq2 6854 . . . . . . . 8 (𝑥 = suc 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝐵))
1211eleq2d 2830 . . . . . . 7 (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴𝑜 𝑥) ↔ 𝐵 ∈ (𝐴𝑜 suc 𝐵)))
1312rspcev 3462 . . . . . 6 ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥))
145, 10, 13syl2anc 579 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥))
15 onintrab2 7204 . . . . 5 (∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥) ↔ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On)
1614, 15sylib 209 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On)
17 onuni 7195 . . . 4 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On)
1816, 17syl 17 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On)
191, 18syl5eqel 2848 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ On)
20 sucidg 5988 . . . . . . 7 (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ suc 𝑋)
22 dif1o 7789 . . . . . . . . . . . . 13 (𝐵 ∈ (On ∖ 1𝑜) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
2322simprbi 490 . . . . . . . . . . . 12 (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ≠ ∅)
2423adantl 473 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ≠ ∅)
25 ssrab2 3849 . . . . . . . . . . . . . . 15 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ⊆ On
26 rabn0 4124 . . . . . . . . . . . . . . . 16 ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥))
2714, 26sylibr 225 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ≠ ∅)
28 onint 7197 . . . . . . . . . . . . . . 15 (({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ⊆ On ∧ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ≠ ∅) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
2925, 27, 28sylancr 581 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
30 eleq1 2832 . . . . . . . . . . . . . 14 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
3129, 30syl5ibcom 236 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
32 oveq2 6854 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
3332eleq2d 2830 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝐵 ∈ (𝐴𝑜 𝑥) ↔ 𝐵 ∈ (𝐴𝑜 ∅)))
3433elrab 3521 . . . . . . . . . . . . . . 15 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴𝑜 ∅)))
3534simprbi 490 . . . . . . . . . . . . . 14 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → 𝐵 ∈ (𝐴𝑜 ∅))
36 eldifi 3896 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
3736adantr 472 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐴 ∈ On)
38 oe0 7811 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
3937, 38syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 ∅) = 1𝑜)
4039eleq2d 2830 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴𝑜 ∅) ↔ 𝐵 ∈ 1𝑜))
41 el1o 7788 . . . . . . . . . . . . . . 15 (𝐵 ∈ 1𝑜𝐵 = ∅)
4240, 41syl6bb 278 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴𝑜 ∅) ↔ 𝐵 = ∅))
4335, 42syl5ib 235 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → 𝐵 = ∅))
4431, 43syld 47 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ → 𝐵 = ∅))
4544necon3ad 2950 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ≠ ∅ → ¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅))
4624, 45mpd 15 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅)
47 limuni 5970 . . . . . . . . . . . . . . . . 17 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
4847, 1syl6eqr 2817 . . . . . . . . . . . . . . . 16 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = 𝑋)
4948adantl 473 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = 𝑋)
5029adantr 472 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
5149, 50eqeltrrd 2845 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
52 oveq2 6854 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑋 → (𝐴𝑜 𝑦) = (𝐴𝑜 𝑋))
5352eleq2d 2830 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑋 → (𝐵 ∈ (𝐴𝑜 𝑦) ↔ 𝐵 ∈ (𝐴𝑜 𝑋)))
54 oveq2 6854 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
5554eleq2d 2830 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝐵 ∈ (𝐴𝑜 𝑥) ↔ 𝐵 ∈ (𝐴𝑜 𝑦)))
5655cbvrabv 3348 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑦)}
5753, 56elrab2 3525 . . . . . . . . . . . . . . 15 (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴𝑜 𝑋)))
5857simprbi 490 . . . . . . . . . . . . . 14 (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → 𝐵 ∈ (𝐴𝑜 𝑋))
5951, 58syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝐵 ∈ (𝐴𝑜 𝑋))
6036ad2antrr 717 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝐴 ∈ On)
61 limeq 5922 . . . . . . . . . . . . . . . . 17 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = 𝑋 → (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ Lim 𝑋))
6248, 61syl 17 . . . . . . . . . . . . . . . 16 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ Lim 𝑋))
6362ibi 258 . . . . . . . . . . . . . . 15 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → Lim 𝑋)
6419, 63anim12i 606 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
65 dif20el 7794 . . . . . . . . . . . . . . 15 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
6665ad2antrr 717 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → ∅ ∈ 𝐴)
67 oelim 7823 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑋) = 𝑦𝑋 (𝐴𝑜 𝑦))
6860, 64, 66, 67syl21anc 866 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → (𝐴𝑜 𝑋) = 𝑦𝑋 (𝐴𝑜 𝑦))
6959, 68eleqtrd 2846 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝐵 𝑦𝑋 (𝐴𝑜 𝑦))
70 eliun 4682 . . . . . . . . . . . 12 (𝐵 𝑦𝑋 (𝐴𝑜 𝑦) ↔ ∃𝑦𝑋 𝐵 ∈ (𝐴𝑜 𝑦))
7169, 70sylib 209 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → ∃𝑦𝑋 𝐵 ∈ (𝐴𝑜 𝑦))
7219adantr 472 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝑋 ∈ On)
73 onss 7192 . . . . . . . . . . . . . . 15 (𝑋 ∈ On → 𝑋 ⊆ On)
7472, 73syl 17 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝑋 ⊆ On)
7574sselda 3763 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) ∧ 𝑦𝑋) → 𝑦 ∈ On)
7649eleq2d 2830 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → (𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ 𝑦𝑋))
7776biimpar 469 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) ∧ 𝑦𝑋) → 𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
7855onnminsb 7206 . . . . . . . . . . . . 13 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → ¬ 𝐵 ∈ (𝐴𝑜 𝑦)))
7975, 77, 78sylc 65 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) ∧ 𝑦𝑋) → ¬ 𝐵 ∈ (𝐴𝑜 𝑦))
8079nrexdv 3147 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → ¬ ∃𝑦𝑋 𝐵 ∈ (𝐴𝑜 𝑦))
8171, 80pm2.65da 851 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
82 ioran 1006 . . . . . . . . . 10 (¬ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) ↔ (¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ ∧ ¬ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
8346, 81, 82sylanbrc 578 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
84 eloni 5920 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
85 unizlim 6026 . . . . . . . . . 10 (Ord {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})))
8616, 84, 853syl 18 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})))
8783, 86mtbird 316 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
88 orduniorsuc 7232 . . . . . . . . . 10 (Ord {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∨ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
8916, 84, 883syl 18 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∨ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
9089ord 890 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
9187, 90mpd 15 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
92 suceq 5975 . . . . . . . 8 (𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → suc 𝑋 = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
931, 92ax-mp 5 . . . . . . 7 suc 𝑋 = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}
9491, 93syl6reqr 2818 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
9521, 94eleqtrd 2846 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
9656inteqi 4639 . . . . 5 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑦)}
9795, 96syl6eleq 2854 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑦)})
9853onnminsb 7206 . . . 4 (𝑋 ∈ On → (𝑋 {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑦)} → ¬ 𝐵 ∈ (𝐴𝑜 𝑋)))
9919, 97, 98sylc 65 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ 𝐵 ∈ (𝐴𝑜 𝑋))
100 oecl 7826 . . . . 5 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
10137, 19, 100syl2anc 579 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 𝑋) ∈ On)
102 ontri1 5944 . . . 4 (((𝐴𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴𝑜 𝑋)))
103101, 3, 102syl2anc 579 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐴𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴𝑜 𝑋)))
10499, 103mpbird 248 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 𝑋) ⊆ 𝐵)
10594, 29eqeltrd 2844 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
106 oveq2 6854 . . . . . 6 (𝑦 = suc 𝑋 → (𝐴𝑜 𝑦) = (𝐴𝑜 suc 𝑋))
107106eleq2d 2830 . . . . 5 (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴𝑜 𝑦) ↔ 𝐵 ∈ (𝐴𝑜 suc 𝑋)))
108107, 56elrab2 3525 . . . 4 (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴𝑜 suc 𝑋)))
109108simprbi 490 . . 3 (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → 𝐵 ∈ (𝐴𝑜 suc 𝑋))
110105, 109syl 17 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴𝑜 suc 𝑋))
11119, 104, 1103jca 1158 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝑋 ∈ On ∧ (𝐴𝑜 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wrex 3056  {crab 3059  cdif 3731  wss 3734  c0 4081   cuni 4596   cint 4635   ciun 4678  Ord word 5909  Oncon0 5910  Lim wlim 5911  suc csuc 5912  (class class class)co 6846  1𝑜c1o 7761  2𝑜c2o 7762  𝑜 coe 7767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-omul 7773  df-oexp 7774
This theorem is referenced by:  oeeui  7891  oeeu  7892
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