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Theorem oawordeulem 8165
 Description: Lemma for oawordex 8168. (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 4008 . . . 4 𝑆 ⊆ On
3 oawordeulem.2 . . . . . 6 𝐵 ∈ On
4 oawordeulem.1 . . . . . . 7 𝐴 ∈ On
5 oaword2 8164 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵))
63, 4, 5mp2an 691 . . . . . 6 𝐵 ⊆ (𝐴 +o 𝐵)
7 oveq2 7143 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
87sseq2d 3947 . . . . . . 7 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
98, 1elrab2 3631 . . . . . 6 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)))
103, 6, 9mpbir2an 710 . . . . 5 𝐵𝑆
1110ne0ii 4253 . . . 4 𝑆 ≠ ∅
12 oninton 7497 . . . 4 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
132, 11, 12mp2an 691 . . 3 𝑆 ∈ On
14 onzsl 7543 . . . . . 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 7143 . . . . . . . . 9 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = (𝐴 +o ∅))
17 oa0 8126 . . . . . . . . . 10 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
184, 17ax-mp 5 . . . . . . . . 9 (𝐴 +o ∅) = 𝐴
1916, 18eqtrdi 2849 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = 𝐴)
2019sseq1d 3946 . . . . . . 7 ( 𝑆 = ∅ → ((𝐴 +o 𝑆) ⊆ 𝐵𝐴𝐵))
2120biimprd 251 . . . . . 6 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
22 oveq2 7143 . . . . . . . . . 10 ( 𝑆 = suc 𝑧 → (𝐴 +o 𝑆) = (𝐴 +o suc 𝑧))
23 oasuc 8134 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
244, 23mpan 689 . . . . . . . . . 10 (𝑧 ∈ On → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
2522, 24sylan9eqr 2855 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) = suc (𝐴 +o 𝑧))
26 vex 3444 . . . . . . . . . . . . 13 𝑧 ∈ V
2726sucid 6238 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
28 eleq2 2878 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
2927, 28mpbiri 261 . . . . . . . . . . 11 ( 𝑆 = suc 𝑧𝑧 𝑆)
3013oneli 6266 . . . . . . . . . . . 12 (𝑧 𝑆𝑧 ∈ On)
311inteqi 4842 . . . . . . . . . . . . . . 15 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
3231eleq2i 2881 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
33 oveq2 7143 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝐴 +o 𝑦) = (𝐴 +o 𝑧))
3433sseq2d 3947 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑧)))
3534onnminsb 7501 . . . . . . . . . . . . . 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 8145 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o 𝑧) ∈ On)
384, 37mpan 689 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝐴 +o 𝑧) ∈ On)
39 ontri1 6193 . . . . . . . . . . . . . . 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 6263 . . . . . . . . . . . 12 Ord 𝐵
45 ordsucss 7515 . . . . . . . . . . . 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 3953 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) ⊆ 𝐵)
5049rexlimiva 3240 . . . . . . 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 8142 . . . . . . . . 9 ((𝐴 ∈ On ∧ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
534, 52mpan 689 . . . . . . . 8 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
54 iunss 4932 . . . . . . . . 9 ( 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵)
553onelssi 6267 . . . . . . . . . 10 ((𝐴 +o 𝑧) ∈ 𝐵 → (𝐴 +o 𝑧) ⊆ 𝐵)
5643, 55syl 17 . . . . . . . . 9 (𝑧 𝑆 → (𝐴 +o 𝑧) ⊆ 𝐵)
5754, 56mprgbir 3121 . . . . . . . 8 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵
5853, 57eqsstrdi 3969 . . . . . . 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 3571 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦))
633, 6, 62mp2an 691 . . . . . 6 𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦)
64 nfcv 2955 . . . . . . . 8 𝑦𝐵
65 nfcv 2955 . . . . . . . . 9 𝑦𝐴
66 nfcv 2955 . . . . . . . . 9 𝑦 +o
67 nfrab1 3337 . . . . . . . . . 10 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6867nfint 4848 . . . . . . . . 9 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6965, 66, 68nfov 7165 . . . . . . . 8 𝑦(𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7064, 69nfss 3907 . . . . . . 7 𝑦 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
71 oveq2 7143 . . . . . . . 8 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7271sseq2d 3947 . . . . . . 7 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
7370, 72onminsb 7496 . . . . . 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 7146 . . . . 5 (𝐴 +o 𝑆) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7674, 75sseqtrri 3952 . . . 4 𝐵 ⊆ (𝐴 +o 𝑆)
77 eqss 3930 . . . 4 ((𝐴 +o 𝑆) = 𝐵 ↔ ((𝐴 +o 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +o 𝑆)))
7861, 76, 77sylanblrc 593 . . 3 (𝐴𝐵 → (𝐴 +o 𝑆) = 𝐵)
79 oveq2 7143 . . . . 5 (𝑥 = 𝑆 → (𝐴 +o 𝑥) = (𝐴 +o 𝑆))
8079eqeq1d 2800 . . . 4 (𝑥 = 𝑆 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑆) = 𝐵))
8180rspcev 3571 . . 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 2820 . . . 4 (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
84 oacan 8159 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
854, 84mp3an1 1445 . . . 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 3168 . 2 𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)
88 oveq2 7143 . . . 4 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
8988eqeq1d 2800 . . 3 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
9089reu4 3670 . 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 1083   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∩ cint 4838  ∪ ciun 4881  Ord word 6158  Oncon0 6159  Lim wlim 6160  suc csuc 6161  (class class class)co 7135   +o coa 8084 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-oadd 8091 This theorem is referenced by:  oawordeu  8166
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