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Theorem oawordeulem 8448
Description: Lemma for oawordex 8451. (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 4026 . . . 4 𝑆 ⊆ On
3 oawordeulem.2 . . . . . 6 𝐵 ∈ On
4 oawordeulem.1 . . . . . . 7 𝐴 ∈ On
5 oaword2 8447 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵))
63, 4, 5mp2an 689 . . . . . 6 𝐵 ⊆ (𝐴 +o 𝐵)
7 oveq2 7337 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
87sseq2d 3963 . . . . . . 7 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
98, 1elrab2 3637 . . . . . 6 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)))
103, 6, 9mpbir2an 708 . . . . 5 𝐵𝑆
1110ne0ii 4283 . . . 4 𝑆 ≠ ∅
12 oninton 7700 . . . 4 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
132, 11, 12mp2an 689 . . 3 𝑆 ∈ On
14 onzsl 7752 . . . . . 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 7337 . . . . . . . . 9 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = (𝐴 +o ∅))
17 oa0 8409 . . . . . . . . . 10 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
184, 17ax-mp 5 . . . . . . . . 9 (𝐴 +o ∅) = 𝐴
1916, 18eqtrdi 2792 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴 +o 𝑆) = 𝐴)
2019sseq1d 3962 . . . . . . 7 ( 𝑆 = ∅ → ((𝐴 +o 𝑆) ⊆ 𝐵𝐴𝐵))
2120biimprd 247 . . . . . 6 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
22 oveq2 7337 . . . . . . . . . 10 ( 𝑆 = suc 𝑧 → (𝐴 +o 𝑆) = (𝐴 +o suc 𝑧))
23 oasuc 8417 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
244, 23mpan 687 . . . . . . . . . 10 (𝑧 ∈ On → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
2522, 24sylan9eqr 2798 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) = suc (𝐴 +o 𝑧))
26 vex 3445 . . . . . . . . . . . . 13 𝑧 ∈ V
2726sucid 6377 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
28 eleq2 2825 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
2927, 28mpbiri 257 . . . . . . . . . . 11 ( 𝑆 = suc 𝑧𝑧 𝑆)
3013oneli 6408 . . . . . . . . . . . 12 (𝑧 𝑆𝑧 ∈ On)
311inteqi 4897 . . . . . . . . . . . . . . 15 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
3231eleq2i 2828 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
33 oveq2 7337 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝐴 +o 𝑦) = (𝐴 +o 𝑧))
3433sseq2d 3963 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑧)))
3534onnminsb 7704 . . . . . . . . . . . . . 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 8428 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o 𝑧) ∈ On)
384, 37mpan 687 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝐴 +o 𝑧) ∈ On)
39 ontri1 6330 . . . . . . . . . . . . . . 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 6405 . . . . . . . . . . . 12 Ord 𝐵
45 ordsucss 7723 . . . . . . . . . . . 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 3969 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +o 𝑆) ⊆ 𝐵)
5049rexlimiva 3140 . . . . . . 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 8425 . . . . . . . . 9 ((𝐴 ∈ On ∧ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
534, 52mpan 687 . . . . . . . 8 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) = 𝑧 𝑆(𝐴 +o 𝑧))
54 iunss 4989 . . . . . . . . 9 ( 𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +o 𝑧) ⊆ 𝐵)
553onelssi 6409 . . . . . . . . . 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 3985 . . . . . . 7 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +o 𝑆) ⊆ 𝐵)
5958a1d 25 . . . . . 6 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴𝐵 → (𝐴 +o 𝑆) ⊆ 𝐵))
6021, 51, 593jaoi 1426 . . . . 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 3570 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦))
633, 6, 62mp2an 689 . . . . . 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 3422 . . . . . . . . . 10 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6867nfint 4903 . . . . . . . . 9 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6965, 66, 68nfov 7359 . . . . . . . 8 𝑦(𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7064, 69nfss 3923 . . . . . . 7 𝑦 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
71 oveq2 7337 . . . . . . . 8 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7271sseq2d 3963 . . . . . . 7 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
7370, 72onminsb 7699 . . . . . 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 7340 . . . . 5 (𝐴 +o 𝑆) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7674, 75sseqtrri 3968 . . . 4 𝐵 ⊆ (𝐴 +o 𝑆)
77 eqss 3946 . . . 4 ((𝐴 +o 𝑆) = 𝐵 ↔ ((𝐴 +o 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +o 𝑆)))
7861, 76, 77sylanblrc 590 . . 3 (𝐴𝐵 → (𝐴 +o 𝑆) = 𝐵)
79 oveq2 7337 . . . . 5 (𝑥 = 𝑆 → (𝐴 +o 𝑥) = (𝐴 +o 𝑆))
8079eqeq1d 2738 . . . 4 (𝑥 = 𝑆 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑆) = 𝐵))
8180rspcev 3570 . . 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 2762 . . . 4 (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
84 oacan 8442 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
854, 84mp3an1 1447 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
8683, 85syl5ib 243 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
8786rgen2 3190 . 2 𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)
88 oveq2 7337 . . . 4 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
8988eqeq1d 2738 . . 3 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
9089reu4 3676 . 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 1085   = wceq 1540  wcel 2105  wne 2940  wral 3061  wrex 3070  ∃!wreu 3347  {crab 3403  Vcvv 3441  wss 3897  c0 4268   cint 4893   ciun 4938  Ord word 6295  Oncon0 6296  Lim wlim 6297  suc csuc 6298  (class class class)co 7329   +o coa 8356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-oadd 8363
This theorem is referenced by:  oawordeu  8449
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