MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeeulem Structured version   Visualization version   GIF version

Theorem oeeulem 8603
Description: Lemma for oeeu 8605. (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 4126 . . . . . . . 8 (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
32adantl 482 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
4 onsuc 7801 . . . . . . 7 (𝐵 ∈ On → suc 𝐵 ∈ On)
53, 4syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ∈ On)
6 oeworde 8595 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴o suc 𝐵))
75, 6syldan 591 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ⊆ (𝐴o suc 𝐵))
8 sucidg 6445 . . . . . . . 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 7419 . . . . . . . 8 (𝑥 = suc 𝐵 → (𝐴o 𝑥) = (𝐴o suc 𝐵))
1211eleq2d 2819 . . . . . . 7 (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴o 𝑥) ↔ 𝐵 ∈ (𝐴o suc 𝐵)))
1312rspcev 3612 . . . . . 6 ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴o suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴o 𝑥))
145, 10, 13syl2anc 584 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴o 𝑥))
15 onintrab2 7787 . . . . 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 7778 . . . 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 2837 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
20 sucidg 6445 . . . . . . 7 (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ suc 𝑋)
22 suceq 6430 . . . . . . . 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 8502 . . . . . . . . . . . . 13 (𝐵 ∈ (On ∖ 1o) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
2524simprbi 497 . . . . . . . . . . . 12 (𝐵 ∈ (On ∖ 1o) → 𝐵 ≠ ∅)
2625adantl 482 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ≠ ∅)
27 ssrab2 4077 . . . . . . . . . . . . . . 15 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ⊆ On
28 rabn0 4385 . . . . . . . . . . . . . . . 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 7780 . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
32 eleq1 2821 . . . . . . . . . . . . . 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 7419 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
3534eleq2d 2819 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝐵 ∈ (𝐴o 𝑥) ↔ 𝐵 ∈ (𝐴o ∅)))
3635elrab 3683 . . . . . . . . . . . . . . 15 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴o ∅)))
3736simprbi 497 . . . . . . . . . . . . . 14 (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → 𝐵 ∈ (𝐴o ∅))
38 eldifi 4126 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
3938adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
40 oe0 8524 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (𝐴o ∅) = 1o)
4139, 40syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o ∅) = 1o)
4241eleq2d 2819 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴o ∅) ↔ 𝐵 ∈ 1o))
43 el1o 8497 . . . . . . . . . . . . . . 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 2953 . . . . . . . . . . 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 6425 . . . . . . . . . . . . . . . . 17 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
5049, 1eqtr4di 2790 . . . . . . . . . . . . . . . 16 (Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = 𝑋)
5150adantl 482 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = 𝑋)
5231adantr 481 . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
54 oveq2 7419 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑋 → (𝐴o 𝑦) = (𝐴o 𝑋))
5554eleq2d 2819 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑋 → (𝐵 ∈ (𝐴o 𝑦) ↔ 𝐵 ∈ (𝐴o 𝑋)))
56 oveq2 7419 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
5756eleq2d 2819 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝐵 ∈ (𝐴o 𝑥) ↔ 𝐵 ∈ (𝐴o 𝑦)))
5857cbvrabv 3442 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑦)}
5955, 58elrab2 3686 . . . . . . . . . . . . . . 15 (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴o 𝑋)))
6059simprbi 497 . . . . . . . . . . . . . 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 6376 . . . . . . . . . . . . . . . . 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 613 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
67 dif20el 8507 . . . . . . . . . . . . . . 15 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
6867ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → ∅ ∈ 𝐴)
69 oelim 8536 . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝐵 𝑦𝑋 (𝐴o 𝑦))
72 eliun 5001 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → 𝑋 ∈ On)
75 onss 7774 . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) → (𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ 𝑦𝑋))
7978biimpar 478 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}) ∧ 𝑦𝑋) → 𝑦 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
8057onnminsb 7789 . . . . . . . . . . . . 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 3149 . . . . . . . . . . 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 982 . . . . . . . . . 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 583 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = ∅ ∨ Lim {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}))
86 eloni 6374 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
87 unizlim 6487 . . . . . . . . . 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 7820 . . . . . . . . . 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 2791 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
9521, 94eleqtrd 2835 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
9658inteqi 4954 . . . . 5 {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑦)}
9795, 96eleqtrdi 2843 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑦)})
9855onnminsb 7789 . . . 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 8539 . . . . 5 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
10139, 19, 100syl2anc 584 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o 𝑋) ∈ On)
102 ontri1 6398 . . . 4 (((𝐴o 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴o 𝑋)))
103101, 3, 102syl2anc 584 . . 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 2833 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)})
106 oveq2 7419 . . . . . 6 (𝑦 = suc 𝑋 → (𝐴o 𝑦) = (𝐴o suc 𝑋))
107106eleq2d 2819 . . . . 5 (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴o 𝑦) ↔ 𝐵 ∈ (𝐴o suc 𝑋)))
108107, 58elrab2 3686 . . . 4 (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴o suc 𝑋)))
109108simprbi 497 . . 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 1128 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wrex 3070  {crab 3432  cdif 3945  wss 3948  c0 4322   cuni 4908   cint 4950   ciun 4997  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7411  1oc1o 8461  2oc2o 8462  o coe 8467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-omul 8473  df-oexp 8474
This theorem is referenced by:  oeeui  8604  oeeu  8605
  Copyright terms: Public domain W3C validator