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Theorem oawordeulem 8282
Description: Lemma for oawordex 8285. (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 3995 . . . 4 𝑆 ⊆ On
3 oawordeulem.2 . . . . . 6 𝐵 ∈ On
4 oawordeulem.1 . . . . . . 7 𝐴 ∈ On
5 oaword2 8281 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵))
63, 4, 5mp2an 692 . . . . . 6 𝐵 ⊆ (𝐴 +o 𝐵)
7 oveq2 7221 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
87sseq2d 3933 . . . . . . 7 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
98, 1elrab2 3605 . . . . . 6 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)))
103, 6, 9mpbir2an 711 . . . . 5 𝐵𝑆
1110ne0ii 4252 . . . 4 𝑆 ≠ ∅
12 oninton 7579 . . . 4 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
132, 11, 12mp2an 692 . . 3 𝑆 ∈ On
14 onzsl 7625 . . . . . 6 ( 𝑆 ∈ On ↔ ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆)))
1513, 14mpbi 233 . . . . 5 ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆))
16 oveq2 7221 . . . . . . . . 9 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = (𝐴 +o ∅))
17 oa0 8243 . . . . . . . . . 10 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
184, 17ax-mp 5 . . . . . . . . 9 (𝐴 +o ∅) = 𝐴
1916, 18eqtrdi 2794 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = 𝐴)
2019sseq1d 3932 . . . . . . 7 ( 𝑆 = ∅ → ((𝐴 +o 𝑆) ⊆ 𝐵𝐴𝐵))
2120biimprd 251 . . . . . 6 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
22 oveq2 7221 . . . . . . . . . 10 ( 𝑆 = suc 𝑧 → (𝐴 +o 𝑆) = (𝐴 +o suc 𝑧))
23 oasuc 8251 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
244, 23mpan 690 . . . . . . . . . 10 (𝑧 ∈ On → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
2522, 24sylan9eqr 2800 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) = suc (𝐴 +o 𝑧))
26 vex 3412 . . . . . . . . . . . . 13 𝑧 ∈ V
2726sucid 6292 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
28 eleq2 2826 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
2927, 28mpbiri 261 . . . . . . . . . . 11 ( 𝑆 = suc 𝑧𝑧 𝑆)
3013oneli 6321 . . . . . . . . . . . 12 (𝑧 𝑆𝑧 ∈ On)
311inteqi 4863 . . . . . . . . . . . . . . 15 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
3231eleq2i 2829 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
33 oveq2 7221 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝐴 +o 𝑦) = (𝐴 +o 𝑧))
3433sseq2d 3933 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑧)))
3534onnminsb 7583 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
3632, 35syl5bi 245 . . . . . . . . . . . . 13 (𝑧 ∈ On → (𝑧 𝑆 → ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
37 oacl 8262 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o 𝑧) ∈ On)
384, 37mpan 690 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝐴 +o 𝑧) ∈ On)
39 ontri1 6247 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ (𝐴 +o 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
403, 38, 39sylancr 590 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
4140con2bid 358 . . . . . . . . . . . . 13 (𝑧 ∈ On → ((𝐴 +o 𝑧) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
4236, 41sylibrd 262 . . . . . . . . . . . 12 (𝑧 ∈ On → (𝑧 𝑆 → (𝐴 +o 𝑧) ∈ 𝐵))
4330, 42mpcom 38 . . . . . . . . . . 11 (𝑧 𝑆 → (𝐴 +o 𝑧) ∈ 𝐵)
443onordi 6318 . . . . . . . . . . . 12 Ord 𝐵
45 ordsucss 7597 . . . . . . . . . . . 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 485 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → suc (𝐴 +o 𝑧) ⊆ 𝐵)
4925, 48eqsstrd 3939 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) ⊆ 𝐵)
5049rexlimiva 3200 . . . . . . 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 8259 . . . . . . . . 9 ((𝐴 ∈ On ∧ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
534, 52mpan 690 . . . . . . . 8 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
54 iunss 4954 . . . . . . . . 9 ( 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵)
553onelssi 6322 . . . . . . . . . 10 ((𝐴 +o 𝑧) ∈ 𝐵 → (𝐴 +o 𝑧) ⊆ 𝐵)
5643, 55syl 17 . . . . . . . . 9 (𝑧 𝑆 → (𝐴 +o 𝑧) ⊆ 𝐵)
5754, 56mprgbir 3076 . . . . . . . 8 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵
5853, 57eqsstrdi 3955 . . . . . . 7 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) ⊆ 𝐵)
5958a1d 25 . . . . . 6 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
6021, 51, 593jaoi 1429 . . . . 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 3537 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦))
633, 6, 62mp2an 692 . . . . . 6 𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦)
64 nfcv 2904 . . . . . . . 8 𝑦𝐵
65 nfcv 2904 . . . . . . . . 9 𝑦𝐴
66 nfcv 2904 . . . . . . . . 9 𝑦 +o
67 nfrab1 3296 . . . . . . . . . 10 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6867nfint 4869 . . . . . . . . 9 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6965, 66, 68nfov 7243 . . . . . . . 8 𝑦(𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7064, 69nfss 3892 . . . . . . 7 𝑦 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
71 oveq2 7221 . . . . . . . 8 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7271sseq2d 3933 . . . . . . 7 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
7370, 72onminsb 7578 . . . . . 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 7224 . . . . 5 (𝐴 +o 𝑆) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7674, 75sseqtrri 3938 . . . 4 𝐵 ⊆ (𝐴 +o 𝑆)
77 eqss 3916 . . . 4 ((𝐴 +o 𝑆) = 𝐵 ↔ ((𝐴 +o 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +o 𝑆)))
7861, 76, 77sylanblrc 593 . . 3 (𝐴𝐵 → (𝐴 +o 𝑆) = 𝐵)
79 oveq2 7221 . . . . 5 (𝑥 = 𝑆 → (𝐴 +o 𝑥) = (𝐴 +o 𝑆))
8079eqeq1d 2739 . . . 4 (𝑥 = 𝑆 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑆) = 𝐵))
8180rspcev 3537 . . 3 (( 𝑆 ∈ On ∧ (𝐴 +o 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
8213, 78, 81sylancr 590 . 2 (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
83 eqtr3 2763 . . . 4 (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
84 oacan 8276 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
854, 84mp3an1 1450 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
8683, 85syl5ib 247 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
8786rgen2 3124 . 2 𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)
88 oveq2 7221 . . . 4 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
8988eqeq1d 2739 . . 3 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
9089reu4 3644 . 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 593 1 (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3o 1088   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  ∃!wreu 3063  {crab 3065  Vcvv 3408  wss 3866  c0 4237   cint 4859   ciun 4904  Ord word 6212  Oncon0 6213  Lim wlim 6214  suc csuc 6215  (class class class)co 7213   +o coa 8199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-oadd 8206
This theorem is referenced by:  oawordeu  8283
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