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Theorem oawordeulem 7580
Description: Lemma for oawordex 7583. (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 . . . . . 6 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
2 ssrab2 3671 . . . . . 6 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
31, 2eqsstri 3619 . . . . 5 𝑆 ⊆ On
4 oawordeulem.2 . . . . . . 7 𝐵 ∈ On
5 oawordeulem.1 . . . . . . . 8 𝐴 ∈ On
6 oaword2 7579 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
74, 5, 6mp2an 707 . . . . . . 7 𝐵 ⊆ (𝐴 +𝑜 𝐵)
8 oveq2 6613 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
98sseq2d 3617 . . . . . . . 8 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
109, 1elrab2 3353 . . . . . . 7 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
114, 7, 10mpbir2an 954 . . . . . 6 𝐵𝑆
1211ne0ii 3904 . . . . 5 𝑆 ≠ ∅
13 oninton 6948 . . . . 5 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
143, 12, 13mp2an 707 . . . 4 𝑆 ∈ On
15 onzsl 6994 . . . . . . . 8 ( 𝑆 ∈ On ↔ ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆)))
1614, 15mpbi 220 . . . . . . 7 ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆))
17 oveq2 6613 . . . . . . . . . . 11 ( 𝑆 = ∅ → (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 ∅))
18 oa0 7542 . . . . . . . . . . . 12 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
195, 18ax-mp 5 . . . . . . . . . . 11 (𝐴 +𝑜 ∅) = 𝐴
2017, 19syl6eq 2676 . . . . . . . . . 10 ( 𝑆 = ∅ → (𝐴 +𝑜 𝑆) = 𝐴)
2120sseq1d 3616 . . . . . . . . 9 ( 𝑆 = ∅ → ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐴𝐵))
2221biimprd 238 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
23 oveq2 6613 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 suc 𝑧))
24 oasuc 7550 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
255, 24mpan 705 . . . . . . . . . . . 12 (𝑧 ∈ On → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
2623, 25sylan9eqr 2682 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +𝑜 𝑆) = suc (𝐴 +𝑜 𝑧))
27 vex 3194 . . . . . . . . . . . . . . 15 𝑧 ∈ V
2827sucid 5766 . . . . . . . . . . . . . 14 𝑧 ∈ suc 𝑧
29 eleq2 2693 . . . . . . . . . . . . . 14 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
3028, 29mpbiri 248 . . . . . . . . . . . . 13 ( 𝑆 = suc 𝑧𝑧 𝑆)
3114oneli 5797 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 ∈ On)
321inteqi 4449 . . . . . . . . . . . . . . . . 17 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
3332eleq2i 2696 . . . . . . . . . . . . . . . 16 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
34 oveq2 6613 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑧))
3534sseq2d 3617 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
3635onnminsb 6952 . . . . . . . . . . . . . . . 16 (𝑧 ∈ On → (𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
3733, 36syl5bi 232 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝑧 𝑆 → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
38 oacl 7561 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 𝑧) ∈ On)
395, 38mpan 705 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ On → (𝐴 +𝑜 𝑧) ∈ On)
40 ontri1 5719 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ (𝐴 +𝑜 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
414, 39, 40sylancr 694 . . . . . . . . . . . . . . . 16 (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
4241con2bid 344 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → ((𝐴 +𝑜 𝑧) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
4337, 42sylibrd 249 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑧 𝑆 → (𝐴 +𝑜 𝑧) ∈ 𝐵))
4431, 43mpcom 38 . . . . . . . . . . . . 13 (𝑧 𝑆 → (𝐴 +𝑜 𝑧) ∈ 𝐵)
454onordi 5794 . . . . . . . . . . . . . 14 Ord 𝐵
46 ordsucss 6966 . . . . . . . . . . . . . 14 (Ord 𝐵 → ((𝐴 +𝑜 𝑧) ∈ 𝐵 → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵))
4745, 46ax-mp 5 . . . . . . . . . . . . 13 ((𝐴 +𝑜 𝑧) ∈ 𝐵 → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
4830, 44, 473syl 18 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
4948adantl 482 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5026, 49eqsstrd 3623 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
5150rexlimiva 3026 . . . . . . . . 9 (∃𝑧 ∈ On 𝑆 = suc 𝑧 → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
5251a1d 25 . . . . . . . 8 (∃𝑧 ∈ On 𝑆 = suc 𝑧 → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
53 oalim 7558 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴 +𝑜 𝑆) = 𝑧 𝑆(𝐴 +𝑜 𝑧))
545, 53mpan 705 . . . . . . . . . 10 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +𝑜 𝑆) = 𝑧 𝑆(𝐴 +𝑜 𝑧))
55 iunss 4532 . . . . . . . . . . 11 ( 𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵)
564onelssi 5798 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝑧) ∈ 𝐵 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5744, 56syl 17 . . . . . . . . . . 11 (𝑧 𝑆 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5855, 57mprgbir 2927 . . . . . . . . . 10 𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵
5954, 58syl6eqss 3639 . . . . . . . . 9 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
6059a1d 25 . . . . . . . 8 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
6122, 52, 603jaoi 1388 . . . . . . 7 (( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
6216, 61ax-mp 5 . . . . . 6 (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
639rspcev 3300 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦))
644, 7, 63mp2an 707 . . . . . . . 8 𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦)
65 nfcv 2767 . . . . . . . . . 10 𝑦𝐵
66 nfcv 2767 . . . . . . . . . . 11 𝑦𝐴
67 nfcv 2767 . . . . . . . . . . 11 𝑦 +𝑜
68 nfrab1 3116 . . . . . . . . . . . 12 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
6968nfint 4456 . . . . . . . . . . 11 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
7066, 67, 69nfov 6631 . . . . . . . . . 10 𝑦(𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7165, 70nfss 3581 . . . . . . . . 9 𝑦 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
72 oveq2 6613 . . . . . . . . . 10 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
7372sseq2d 3617 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
7471, 73onminsb 6947 . . . . . . . 8 (∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦) → 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
7564, 74ax-mp 5 . . . . . . 7 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7632oveq2i 6616 . . . . . . 7 (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7775, 76sseqtr4i 3622 . . . . . 6 𝐵 ⊆ (𝐴 +𝑜 𝑆)
7862, 77jctir 560 . . . . 5 (𝐴𝐵 → ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +𝑜 𝑆)))
79 eqss 3603 . . . . 5 ((𝐴 +𝑜 𝑆) = 𝐵 ↔ ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +𝑜 𝑆)))
8078, 79sylibr 224 . . . 4 (𝐴𝐵 → (𝐴 +𝑜 𝑆) = 𝐵)
81 oveq2 6613 . . . . . 6 (𝑥 = 𝑆 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑆))
8281eqeq1d 2628 . . . . 5 (𝑥 = 𝑆 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑆) = 𝐵))
8382rspcev 3300 . . . 4 (( 𝑆 ∈ On ∧ (𝐴 +𝑜 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
8414, 80, 83sylancr 694 . . 3 (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
85 eqtr3 2647 . . . . 5 (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
86 oacan 7574 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦) ↔ 𝑥 = 𝑦))
875, 86mp3an1 1408 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦) ↔ 𝑥 = 𝑦))
8885, 87syl5ib 234 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦))
8988rgen2a 2976 . . 3 𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦)
9084, 89jctir 560 . 2 (𝐴𝐵 → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦)))
91 oveq2 6613 . . . 4 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
9291eqeq1d 2628 . . 3 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑦) = 𝐵))
9392reu4 3387 . 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)))
9490, 93sylibr 224 1 (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  ∃!wreu 2914  {crab 2916  Vcvv 3191  wss 3560  c0 3896   cint 4445   ciun 4490  Ord word 5684  Oncon0 5685  Lim wlim 5686  suc csuc 5687  (class class class)co 6605   +𝑜 coa 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510
This theorem is referenced by:  oawordeu  7581
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