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Theorem oeeulem 8628
Description: Lemma for oeeu 8630. (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 4127 . . . . . . . 8 (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
32adantl 480 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
4 onsuc 7820 . . . . . . 7 (𝐵 ∈ On → suc 𝐵 ∈ On)
53, 4syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ∈ On)
6 oeworde 8620 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴o suc 𝐵))
75, 6syldan 589 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ⊆ (𝐴o suc 𝐵))
8 sucidg 6455 . . . . . . . 8 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
93, 8syl 17 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ suc 𝐵)
107, 9sseldd 3983 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ (𝐴o suc 𝐵))
11 oveq2 7434 . . . . . . . 8 (𝑥 = suc 𝐵 → (𝐴o 𝑥) = (𝐴o suc 𝐵))
1211eleq2d 2815 . . . . . . 7 (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴o 𝑥) ↔ 𝐵 ∈ (𝐴o suc 𝐵)))
1312rspcev 3611 . . . . . 6 ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴o suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴o 𝑥))
145, 10, 13syl2anc 582 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴o 𝑥))
15 onintrab2 7806 . . . . 5 (∃𝑥 ∈ On 𝐵 ∈ (𝐴o 𝑥) ↔ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ On)
1614, 15sylib 217 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ On)
17 onuni 7797 . . . 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, 18eqeltrid 2833 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
20 sucidg 6455 . . . . . . 7 (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ suc 𝑋)
22 suceq 6440 . . . . . . . 8 (𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → suc 𝑋 = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
231, 22ax-mp 5 . . . . . . 7 suc 𝑋 = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}
24 dif1o 8527 . . . . . . . . . . . . 13 (𝐵 ∈ (On ∖ 1o) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
2524simprbi 495 . . . . . . . . . . . 12 (𝐵 ∈ (On ∖ 1o) → 𝐵 ≠ ∅)
2625adantl 480 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ≠ ∅)
27 ssrab2 4077 . . . . . . . . . . . . . . 15 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ⊆ On
28 rabn0 4389 . . . . . . . . . . . . . . . 16 ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴o 𝑥))
2914, 28sylibr 233 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ≠ ∅)
30 onint 7799 . . . . . . . . . . . . . . 15 (({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ⊆ On ∧ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ≠ ∅) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
3127, 29, 30sylancr 585 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
32 eleq1 2817 . . . . . . . . . . . . . 14 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅ → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}))
3331, 32syl5ibcom 244 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}))
34 oveq2 7434 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
3534eleq2d 2815 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝐵 ∈ (𝐴o 𝑥) ↔ 𝐵 ∈ (𝐴o ∅)))
3635elrab 3684 . . . . . . . . . . . . . . 15 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴o ∅)))
3736simprbi 495 . . . . . . . . . . . . . 14 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → 𝐵 ∈ (𝐴o ∅))
38 eldifi 4127 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
3938adantr 479 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
40 oe0 8549 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (𝐴o ∅) = 1o)
4139, 40syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o ∅) = 1o)
4241eleq2d 2815 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴o ∅) ↔ 𝐵 ∈ 1o))
43 el1o 8522 . . . . . . . . . . . . . . 15 (𝐵 ∈ 1o𝐵 = ∅)
4442, 43bitrdi 286 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴o ∅) ↔ 𝐵 = ∅))
4537, 44imbitrid 243 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → 𝐵 = ∅))
4633, 45syld 47 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅ → 𝐵 = ∅))
4746necon3ad 2950 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ≠ ∅ → ¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅))
4826, 47mpd 15 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅)
49 limuni 6435 . . . . . . . . . . . . . . . . 17 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
5049, 1eqtr4di 2786 . . . . . . . . . . . . . . . 16 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = 𝑋)
5150adantl 480 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = 𝑋)
5231adantr 479 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
5351, 52eqeltrrd 2830 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
54 oveq2 7434 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑋 → (𝐴o 𝑦) = (𝐴o 𝑋))
5554eleq2d 2815 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑋 → (𝐵 ∈ (𝐴o 𝑦) ↔ 𝐵 ∈ (𝐴o 𝑋)))
56 oveq2 7434 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
5756eleq2d 2815 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝐵 ∈ (𝐴o 𝑥) ↔ 𝐵 ∈ (𝐴o 𝑦)))
5857cbvrabv 3441 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑦)}
5955, 58elrab2 3687 . . . . . . . . . . . . . . 15 (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴o 𝑋)))
6059simprbi 495 . . . . . . . . . . . . . 14 (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → 𝐵 ∈ (𝐴o 𝑋))
6153, 60syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝐵 ∈ (𝐴o 𝑋))
6238ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝐴 ∈ On)
63 limeq 6386 . . . . . . . . . . . . . . . . 17 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = 𝑋 → (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ Lim 𝑋))
6450, 63syl 17 . . . . . . . . . . . . . . . 16 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ Lim 𝑋))
6564ibi 266 . . . . . . . . . . . . . . 15 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → Lim 𝑋)
6619, 65anim12i 611 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
67 dif20el 8532 . . . . . . . . . . . . . . 15 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
6867ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → ∅ ∈ 𝐴)
69 oelim 8561 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑋) = 𝑦𝑋 (𝐴o 𝑦))
7062, 66, 68, 69syl21anc 836 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → (𝐴o 𝑋) = 𝑦𝑋 (𝐴o 𝑦))
7161, 70eleqtrd 2831 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝐵 𝑦𝑋 (𝐴o 𝑦))
72 eliun 5004 . . . . . . . . . . . 12 (𝐵 𝑦𝑋 (𝐴o 𝑦) ↔ ∃𝑦𝑋 𝐵 ∈ (𝐴o 𝑦))
7371, 72sylib 217 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → ∃𝑦𝑋 𝐵 ∈ (𝐴o 𝑦))
7419adantr 479 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝑋 ∈ On)
75 onss 7793 . . . . . . . . . . . . . . 15 (𝑋 ∈ On → 𝑋 ⊆ On)
7674, 75syl 17 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝑋 ⊆ On)
7776sselda 3982 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) ∧ 𝑦𝑋) → 𝑦 ∈ On)
7851eleq2d 2815 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → (𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ 𝑦𝑋))
7978biimpar 476 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) ∧ 𝑦𝑋) → 𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
8057onnminsb 7808 . . . . . . . . . . . . 13 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → ¬ 𝐵 ∈ (𝐴o 𝑦)))
8177, 79, 80sylc 65 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) ∧ 𝑦𝑋) → ¬ 𝐵 ∈ (𝐴o 𝑦))
8281nrexdv 3146 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → ¬ ∃𝑦𝑋 𝐵 ∈ (𝐴o 𝑦))
8373, 82pm2.65da 815 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
84 ioran 981 . . . . . . . . . 10 (¬ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) ↔ (¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅ ∧ ¬ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}))
8548, 83, 84sylanbrc 581 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}))
86 eloni 6384 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
87 unizlim 6497 . . . . . . . . . 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)})))
8816, 86, 873syl 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)})))
8985, 88mtbird 324 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
90 orduniorsuc 7839 . . . . . . . . . 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)}))
9116, 86, 903syl 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)}))
9291ord 862 . . . . . . . 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)}))
9389, 92mpd 15 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = suc {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
9423, 93eqtr4id 2787 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
9521, 94eleqtrd 2831 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
9658inteqi 4957 . . . . 5 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑦)}
9795, 96eleqtrdi 2839 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑦)})
9855onnminsb 7808 . . . 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 8564 . . . . 5 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
10139, 19, 100syl2anc 582 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o 𝑋) ∈ On)
102 ontri1 6408 . . . 4 (((𝐴o 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴o 𝑋)))
103101, 3, 102syl2anc 582 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ((𝐴o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴o 𝑋)))
10499, 103mpbird 256 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o 𝑋) ⊆ 𝐵)
10594, 31eqeltrd 2829 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
106 oveq2 7434 . . . . . 6 (𝑦 = suc 𝑋 → (𝐴o 𝑦) = (𝐴o suc 𝑋))
107106eleq2d 2815 . . . . 5 (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴o 𝑦) ↔ 𝐵 ∈ (𝐴o suc 𝑋)))
108107, 58elrab2 3687 . . . 4 (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴o suc 𝑋)))
109108simprbi 495 . . 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 1125 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 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wrex 3067  {crab 3430  cdif 3946  wss 3949  c0 4326   cuni 4912   cint 4953   ciun 5000  Ord word 6373  Oncon0 6374  Lim wlim 6375  suc csuc 6376  (class class class)co 7426  1oc1o 8486  2oc2o 8487  o coe 8492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-oadd 8497  df-omul 8498  df-oexp 8499
This theorem is referenced by:  oeeui  8629  oeeu  8630
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