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Theorem oawordeulem 8575
Description: Lemma for oawordex 8578. (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 4076 . . . 4 𝑆 ⊆ On
3 oawordeulem.2 . . . . . 6 𝐵 ∈ On
4 oawordeulem.1 . . . . . . 7 𝐴 ∈ On
5 oaword2 8574 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵))
63, 4, 5mp2an 690 . . . . . 6 𝐵 ⊆ (𝐴 +o 𝐵)
7 oveq2 7427 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
87sseq2d 4009 . . . . . . 7 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
98, 1elrab2 3682 . . . . . 6 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)))
103, 6, 9mpbir2an 709 . . . . 5 𝐵𝑆
1110ne0ii 4337 . . . 4 𝑆 ≠ ∅
12 oninton 7799 . . . 4 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
132, 11, 12mp2an 690 . . 3 𝑆 ∈ On
14 onzsl 7851 . . . . . 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 7427 . . . . . . . . 9 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = (𝐴 +o ∅))
17 oa0 8537 . . . . . . . . . 10 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
184, 17ax-mp 5 . . . . . . . . 9 (𝐴 +o ∅) = 𝐴
1916, 18eqtrdi 2781 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = 𝐴)
2019sseq1d 4008 . . . . . . 7 ( 𝑆 = ∅ → ((𝐴 +o 𝑆) ⊆ 𝐵𝐴𝐵))
2120biimprd 247 . . . . . 6 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
22 oveq2 7427 . . . . . . . . . 10 ( 𝑆 = suc 𝑧 → (𝐴 +o 𝑆) = (𝐴 +o suc 𝑧))
23 oasuc 8545 . . . . . . . . . . 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 2787 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) = suc (𝐴 +o 𝑧))
26 vex 3465 . . . . . . . . . . . . 13 𝑧 ∈ V
2726sucid 6453 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
28 eleq2 2814 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
2927, 28mpbiri 257 . . . . . . . . . . 11 ( 𝑆 = suc 𝑧𝑧 𝑆)
3013oneli 6485 . . . . . . . . . . . 12 (𝑧 𝑆𝑧 ∈ On)
311inteqi 4954 . . . . . . . . . . . . . . 15 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
3231eleq2i 2817 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
33 oveq2 7427 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝐴 +o 𝑦) = (𝐴 +o 𝑧))
3433sseq2d 4009 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑧)))
3534onnminsb 7803 . . . . . . . . . . . . . 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 8556 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o 𝑧) ∈ On)
384, 37mpan 688 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝐴 +o 𝑧) ∈ On)
39 ontri1 6405 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ (𝐴 +o 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
403, 38, 39sylancr 585 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
4140con2bid 353 . . . . . . . . . . . . 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 6482 . . . . . . . . . . . 12 Ord 𝐵
45 ordsucss 7822 . . . . . . . . . . . 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 480 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → suc (𝐴 +o 𝑧) ⊆ 𝐵)
4925, 48eqsstrd 4015 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) ⊆ 𝐵)
5049rexlimiva 3136 . . . . . . 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 8553 . . . . . . . . 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 5049 . . . . . . . . 9 ( 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵)
553onelssi 6486 . . . . . . . . . 10 ((𝐴 +o 𝑧) ∈ 𝐵 → (𝐴 +o 𝑧) ⊆ 𝐵)
5643, 55syl 17 . . . . . . . . 9 (𝑧 𝑆 → (𝐴 +o 𝑧) ⊆ 𝐵)
5754, 56mprgbir 3057 . . . . . . . 8 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵
5853, 57eqsstrdi 4031 . . . . . . 7 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) ⊆ 𝐵)
5958a1d 25 . . . . . 6 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
6021, 51, 593jaoi 1424 . . . . 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 3606 . . . . . . 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 2891 . . . . . . . 8 𝑦𝐵
65 nfcv 2891 . . . . . . . . 9 𝑦𝐴
66 nfcv 2891 . . . . . . . . 9 𝑦 +o
67 nfrab1 3438 . . . . . . . . . 10 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6867nfint 4960 . . . . . . . . 9 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6965, 66, 68nfov 7449 . . . . . . . 8 𝑦(𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7064, 69nfss 3969 . . . . . . 7 𝑦 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
71 oveq2 7427 . . . . . . . 8 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7271sseq2d 4009 . . . . . . 7 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
7370, 72onminsb 7798 . . . . . 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 7430 . . . . 5 (𝐴 +o 𝑆) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7674, 75sseqtrri 4014 . . . 4 𝐵 ⊆ (𝐴 +o 𝑆)
77 eqss 3992 . . . 4 ((𝐴 +o 𝑆) = 𝐵 ↔ ((𝐴 +o 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +o 𝑆)))
7861, 76, 77sylanblrc 588 . . 3 (𝐴𝐵 → (𝐴 +o 𝑆) = 𝐵)
79 oveq2 7427 . . . . 5 (𝑥 = 𝑆 → (𝐴 +o 𝑥) = (𝐴 +o 𝑆))
8079eqeq1d 2727 . . . 4 (𝑥 = 𝑆 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑆) = 𝐵))
8180rspcev 3606 . . 3 (( 𝑆 ∈ On ∧ (𝐴 +o 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
8213, 78, 81sylancr 585 . 2 (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
83 eqtr3 2751 . . . 4 (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
84 oacan 8569 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
854, 84mp3an1 1444 . . . 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 3187 . 2 𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)
88 oveq2 7427 . . . 4 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
8988eqeq1d 2727 . . 3 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
9089reu4 3723 . 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 588 1 (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3o 1083   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  ∃!wreu 3361  {crab 3418  Vcvv 3461  wss 3944  c0 4322   cint 4950   ciun 4997  Ord word 6370  Oncon0 6371  Lim wlim 6372  suc csuc 6373  (class class class)co 7419   +o coa 8484
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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-oadd 8491
This theorem is referenced by:  oawordeu  8576
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