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Theorem oawordeulem 8553
Description: Lemma for oawordex 8556. (Contributed by NM, 11-Dec-2004.)
Hypotheses
Ref Expression
oawordeulem.1 𝐴 ∈ On
oawordeulem.2 𝐵 ∈ On
oawordeulem.3 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
Assertion
Ref Expression
oawordeulem (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem oawordeulem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 oawordeulem.3 . . . . 5 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
21ssrab3 4080 . . . 4 𝑆 ⊆ On
3 oawordeulem.2 . . . . . 6 𝐵 ∈ On
4 oawordeulem.1 . . . . . . 7 𝐴 ∈ On
5 oaword2 8552 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵))
63, 4, 5mp2an 690 . . . . . 6 𝐵 ⊆ (𝐴 +o 𝐵)
7 oveq2 7416 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
87sseq2d 4014 . . . . . . 7 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
98, 1elrab2 3686 . . . . . 6 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)))
103, 6, 9mpbir2an 709 . . . . 5 𝐵𝑆
1110ne0ii 4337 . . . 4 𝑆 ≠ ∅
12 oninton 7782 . . . 4 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
132, 11, 12mp2an 690 . . 3 𝑆 ∈ On
14 onzsl 7834 . . . . . 6 ( 𝑆 ∈ On ↔ ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆)))
1513, 14mpbi 229 . . . . 5 ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆))
16 oveq2 7416 . . . . . . . . 9 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = (𝐴 +o ∅))
17 oa0 8515 . . . . . . . . . 10 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
184, 17ax-mp 5 . . . . . . . . 9 (𝐴 +o ∅) = 𝐴
1916, 18eqtrdi 2788 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = 𝐴)
2019sseq1d 4013 . . . . . . 7 ( 𝑆 = ∅ → ((𝐴 +o 𝑆) ⊆ 𝐵𝐴𝐵))
2120biimprd 247 . . . . . 6 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
22 oveq2 7416 . . . . . . . . . 10 ( 𝑆 = suc 𝑧 → (𝐴 +o 𝑆) = (𝐴 +o suc 𝑧))
23 oasuc 8523 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
244, 23mpan 688 . . . . . . . . . 10 (𝑧 ∈ On → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
2522, 24sylan9eqr 2794 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) = suc (𝐴 +o 𝑧))
26 vex 3478 . . . . . . . . . . . . 13 𝑧 ∈ V
2726sucid 6446 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
28 eleq2 2822 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
2927, 28mpbiri 257 . . . . . . . . . . 11 ( 𝑆 = suc 𝑧𝑧 𝑆)
3013oneli 6478 . . . . . . . . . . . 12 (𝑧 𝑆𝑧 ∈ On)
311inteqi 4954 . . . . . . . . . . . . . . 15 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
3231eleq2i 2825 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
33 oveq2 7416 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝐴 +o 𝑦) = (𝐴 +o 𝑧))
3433sseq2d 4014 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑧)))
3534onnminsb 7786 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
3632, 35biimtrid 241 . . . . . . . . . . . . 13 (𝑧 ∈ On → (𝑧 𝑆 → ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
37 oacl 8534 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o 𝑧) ∈ On)
384, 37mpan 688 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝐴 +o 𝑧) ∈ On)
39 ontri1 6398 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ (𝐴 +o 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
403, 38, 39sylancr 587 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
4140con2bid 354 . . . . . . . . . . . . 13 (𝑧 ∈ On → ((𝐴 +o 𝑧) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
4236, 41sylibrd 258 . . . . . . . . . . . 12 (𝑧 ∈ On → (𝑧 𝑆 → (𝐴 +o 𝑧) ∈ 𝐵))
4330, 42mpcom 38 . . . . . . . . . . 11 (𝑧 𝑆 → (𝐴 +o 𝑧) ∈ 𝐵)
443onordi 6475 . . . . . . . . . . . 12 Ord 𝐵
45 ordsucss 7805 . . . . . . . . . . . 12 (Ord 𝐵 → ((𝐴 +o 𝑧) ∈ 𝐵 → suc (𝐴 +o 𝑧) ⊆ 𝐵))
4644, 45ax-mp 5 . . . . . . . . . . 11 ((𝐴 +o 𝑧) ∈ 𝐵 → suc (𝐴 +o 𝑧) ⊆ 𝐵)
4729, 43, 463syl 18 . . . . . . . . . 10 ( 𝑆 = suc 𝑧 → suc (𝐴 +o 𝑧) ⊆ 𝐵)
4847adantl 482 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → suc (𝐴 +o 𝑧) ⊆ 𝐵)
4925, 48eqsstrd 4020 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) ⊆ 𝐵)
5049rexlimiva 3147 . . . . . . 7 (∃𝑧 ∈ On 𝑆 = suc 𝑧 → (𝐴 +o 𝑆) ⊆ 𝐵)
5150a1d 25 . . . . . 6 (∃𝑧 ∈ On 𝑆 = suc 𝑧 → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
52 oalim 8531 . . . . . . . . 9 ((𝐴 ∈ On ∧ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
534, 52mpan 688 . . . . . . . 8 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
54 iunss 5048 . . . . . . . . 9 ( 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵)
553onelssi 6479 . . . . . . . . . 10 ((𝐴 +o 𝑧) ∈ 𝐵 → (𝐴 +o 𝑧) ⊆ 𝐵)
5643, 55syl 17 . . . . . . . . 9 (𝑧 𝑆 → (𝐴 +o 𝑧) ⊆ 𝐵)
5754, 56mprgbir 3068 . . . . . . . 8 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵
5853, 57eqsstrdi 4036 . . . . . . 7 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) ⊆ 𝐵)
5958a1d 25 . . . . . 6 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
6021, 51, 593jaoi 1427 . . . . 5 (( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
6115, 60ax-mp 5 . . . 4 (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵)
628rspcev 3612 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦))
633, 6, 62mp2an 690 . . . . . 6 𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦)
64 nfcv 2903 . . . . . . . 8 𝑦𝐵
65 nfcv 2903 . . . . . . . . 9 𝑦𝐴
66 nfcv 2903 . . . . . . . . 9 𝑦 +o
67 nfrab1 3451 . . . . . . . . . 10 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6867nfint 4960 . . . . . . . . 9 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6965, 66, 68nfov 7438 . . . . . . . 8 𝑦(𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7064, 69nfss 3974 . . . . . . 7 𝑦 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
71 oveq2 7416 . . . . . . . 8 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7271sseq2d 4014 . . . . . . 7 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
7370, 72onminsb 7781 . . . . . 6 (∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦) → 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7463, 73ax-mp 5 . . . . 5 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7531oveq2i 7419 . . . . 5 (𝐴 +o 𝑆) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7674, 75sseqtrri 4019 . . . 4 𝐵 ⊆ (𝐴 +o 𝑆)
77 eqss 3997 . . . 4 ((𝐴 +o 𝑆) = 𝐵 ↔ ((𝐴 +o 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +o 𝑆)))
7861, 76, 77sylanblrc 590 . . 3 (𝐴𝐵 → (𝐴 +o 𝑆) = 𝐵)
79 oveq2 7416 . . . . 5 (𝑥 = 𝑆 → (𝐴 +o 𝑥) = (𝐴 +o 𝑆))
8079eqeq1d 2734 . . . 4 (𝑥 = 𝑆 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑆) = 𝐵))
8180rspcev 3612 . . 3 (( 𝑆 ∈ On ∧ (𝐴 +o 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
8213, 78, 81sylancr 587 . 2 (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
83 eqtr3 2758 . . . 4 (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
84 oacan 8547 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
854, 84mp3an1 1448 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
8683, 85imbitrid 243 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
8786rgen2 3197 . 2 𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)
88 oveq2 7416 . . . 4 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
8988eqeq1d 2734 . . 3 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
9089reu4 3727 . 2 (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 ↔ (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)))
9182, 87, 90sylanblrc 590 1 (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3o 1086   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  ∃!wreu 3374  {crab 3432  Vcvv 3474  wss 3948  c0 4322   cint 4950   ciun 4997  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7408   +o coa 8462
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 7724
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-rmo 3376  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 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-oadd 8469
This theorem is referenced by:  oawordeu  8554
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