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Theorem oeeulem 7633
Description: Lemma for oeeu 7635. (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 3715 . . . . . . . 8 (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ∈ On)
32adantl 482 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ On)
4 suceloni 6967 . . . . . . 7 (𝐵 ∈ On → suc 𝐵 ∈ On)
53, 4syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ∈ On)
6 oeworde 7625 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴𝑜 suc 𝐵))
75, 6syldan 487 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ⊆ (𝐴𝑜 suc 𝐵))
8 sucidg 5767 . . . . . . . 8 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
93, 8syl 17 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ suc 𝐵)
107, 9sseldd 3588 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴𝑜 suc 𝐵))
11 oveq2 6618 . . . . . . . 8 (𝑥 = suc 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝐵))
1211eleq2d 2684 . . . . . . 7 (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴𝑜 𝑥) ↔ 𝐵 ∈ (𝐴𝑜 suc 𝐵)))
1312rspcev 3298 . . . . . 6 ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥))
145, 10, 13syl2anc 692 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥))
15 onintrab2 6956 . . . . 5 (∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥) ↔ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On)
1614, 15sylib 208 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On)
17 onuni 6947 . . . 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 2702 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ On)
20 sucidg 5767 . . . . . . 7 (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ suc 𝑋)
22 dif1o 7532 . . . . . . . . . . . . 13 (𝐵 ∈ (On ∖ 1𝑜) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
2322simprbi 480 . . . . . . . . . . . 12 (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ≠ ∅)
2423adantl 482 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ≠ ∅)
25 ssrab2 3671 . . . . . . . . . . . . . . 15 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ⊆ On
26 rabn0 3937 . . . . . . . . . . . . . . . 16 ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴𝑜 𝑥))
2714, 26sylibr 224 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ≠ ∅)
28 onint 6949 . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
30 eleq1 2686 . . . . . . . . . . . . . 14 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
3129, 30syl5ibcom 235 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
32 oveq2 6618 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
3332eleq2d 2684 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝐵 ∈ (𝐴𝑜 𝑥) ↔ 𝐵 ∈ (𝐴𝑜 ∅)))
3433elrab 3350 . . . . . . . . . . . . . . 15 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴𝑜 ∅)))
3534simprbi 480 . . . . . . . . . . . . . 14 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → 𝐵 ∈ (𝐴𝑜 ∅))
36 eldifi 3715 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
3736adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐴 ∈ On)
38 oe0 7554 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
3937, 38syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 ∅) = 1𝑜)
4039eleq2d 2684 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴𝑜 ∅) ↔ 𝐵 ∈ 1𝑜))
41 el1o 7531 . . . . . . . . . . . . . . 15 (𝐵 ∈ 1𝑜𝐵 = ∅)
4240, 41syl6bb 276 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴𝑜 ∅) ↔ 𝐵 = ∅))
4335, 42syl5ib 234 . . . . . . . . . . . . 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 2803 . . . . . . . . . . 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 5749 . . . . . . . . . . . . . . . . 17 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
4847, 1syl6eqr 2673 . . . . . . . . . . . . . . . 16 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = 𝑋)
4948adantl 482 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = 𝑋)
5029adantr 481 . . . . . . . . . . . . . . 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 2699 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
52 oveq2 6618 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑋 → (𝐴𝑜 𝑦) = (𝐴𝑜 𝑋))
5352eleq2d 2684 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑋 → (𝐵 ∈ (𝐴𝑜 𝑦) ↔ 𝐵 ∈ (𝐴𝑜 𝑋)))
54 oveq2 6618 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
5554eleq2d 2684 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝐵 ∈ (𝐴𝑜 𝑥) ↔ 𝐵 ∈ (𝐴𝑜 𝑦)))
5655cbvrabv 3188 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑦)}
5753, 56elrab2 3352 . . . . . . . . . . . . . . 15 (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴𝑜 𝑋)))
5857simprbi 480 . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝐴 ∈ On)
61 limeq 5699 . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . 15 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} → Lim 𝑋)
6419, 63anim12i 589 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
65 dif20el 7537 . . . . . . . . . . . . . . 15 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
6665ad2antrr 761 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → ∅ ∈ 𝐴)
67 oelim 7566 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑋) = 𝑦𝑋 (𝐴𝑜 𝑦))
6860, 64, 66, 67syl21anc 1322 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → (𝐴𝑜 𝑋) = 𝑦𝑋 (𝐴𝑜 𝑦))
6959, 68eleqtrd 2700 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝐵 𝑦𝑋 (𝐴𝑜 𝑦))
70 eliun 4495 . . . . . . . . . . . 12 (𝐵 𝑦𝑋 (𝐴𝑜 𝑦) ↔ ∃𝑦𝑋 𝐵 ∈ (𝐴𝑜 𝑦))
7169, 70sylib 208 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → ∃𝑦𝑋 𝐵 ∈ (𝐴𝑜 𝑦))
7219adantr 481 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝑋 ∈ On)
73 onss 6944 . . . . . . . . . . . . . . 15 (𝑋 ∈ On → 𝑋 ⊆ On)
7472, 73syl 17 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → 𝑋 ⊆ On)
7574sselda 3587 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) ∧ 𝑦𝑋) → 𝑦 ∈ On)
7649eleq2d 2684 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → (𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ 𝑦𝑋))
7776biimpar 502 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) ∧ 𝑦𝑋) → 𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
7855onnminsb 6958 . . . . . . . . . . . . 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 2996 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}) → ¬ ∃𝑦𝑋 𝐵 ∈ (𝐴𝑜 𝑦))
8171, 80pm2.65da 599 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
82 ioran 511 . . . . . . . . . 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 697 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}))
84 eloni 5697 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
85 unizlim 5808 . . . . . . . . . 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 315 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
88 orduniorsuc 6984 . . . . . . . . . 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 392 . . . . . . . 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 5754 . . . . . . . 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 2674 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
9521, 94eleqtrd 2700 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
9656inteqi 4449 . . . . 5 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑦)}
9795, 96syl6eleq 2708 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑦)})
9853onnminsb 6958 . . . 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 7569 . . . . 5 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
10137, 19, 100syl2anc 692 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 𝑋) ∈ On)
102 ontri1 5721 . . . 4 (((𝐴𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴𝑜 𝑋)))
103101, 3, 102syl2anc 692 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐴𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴𝑜 𝑋)))
10499, 103mpbird 247 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴𝑜 𝑋) ⊆ 𝐵)
10594, 29eqeltrd 2698 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)})
106 oveq2 6618 . . . . . 6 (𝑦 = suc 𝑋 → (𝐴𝑜 𝑦) = (𝐴𝑜 suc 𝑋))
107106eleq2d 2684 . . . . 5 (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴𝑜 𝑦) ↔ 𝐵 ∈ (𝐴𝑜 suc 𝑋)))
108107, 56elrab2 3352 . . . 4 (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴𝑜 suc 𝑋)))
109108simprbi 480 . . 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 1240 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 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  cdif 3556  wss 3559  c0 3896   cuni 4407   cint 4445   ciun 4490  Ord word 5686  Oncon0 5687  Lim wlim 5688  suc csuc 5689  (class class class)co 6610  1𝑜c1o 7505  2𝑜c2o 7506  𝑜 coe 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-oexp 7518
This theorem is referenced by:  oeeui  7634  oeeu  7635
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