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Theorem oawordeulem 7679
 Description: Lemma for oawordex 7682. (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 3720 . . . . . 6 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
31, 2eqsstri 3668 . . . . 5 𝑆 ⊆ On
4 oawordeulem.2 . . . . . . 7 𝐵 ∈ On
5 oawordeulem.1 . . . . . . . 8 𝐴 ∈ On
6 oaword2 7678 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
74, 5, 6mp2an 708 . . . . . . 7 𝐵 ⊆ (𝐴 +𝑜 𝐵)
8 oveq2 6698 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
98sseq2d 3666 . . . . . . . 8 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
109, 1elrab2 3399 . . . . . . 7 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
114, 7, 10mpbir2an 975 . . . . . 6 𝐵𝑆
1211ne0ii 3956 . . . . 5 𝑆 ≠ ∅
13 oninton 7042 . . . . 5 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
143, 12, 13mp2an 708 . . . 4 𝑆 ∈ On
15 onzsl 7088 . . . . . . . 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 6698 . . . . . . . . . . 11 ( 𝑆 = ∅ → (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 ∅))
18 oa0 7641 . . . . . . . . . . . 12 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
195, 18ax-mp 5 . . . . . . . . . . 11 (𝐴 +𝑜 ∅) = 𝐴
2017, 19syl6eq 2701 . . . . . . . . . 10 ( 𝑆 = ∅ → (𝐴 +𝑜 𝑆) = 𝐴)
2120sseq1d 3665 . . . . . . . . 9 ( 𝑆 = ∅ → ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐴𝐵))
2221biimprd 238 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
23 oveq2 6698 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 suc 𝑧))
24 oasuc 7649 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
255, 24mpan 706 . . . . . . . . . . . 12 (𝑧 ∈ On → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
2623, 25sylan9eqr 2707 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +𝑜 𝑆) = suc (𝐴 +𝑜 𝑧))
27 vex 3234 . . . . . . . . . . . . . . 15 𝑧 ∈ V
2827sucid 5842 . . . . . . . . . . . . . 14 𝑧 ∈ suc 𝑧
29 eleq2 2719 . . . . . . . . . . . . . 14 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
3028, 29mpbiri 248 . . . . . . . . . . . . 13 ( 𝑆 = suc 𝑧𝑧 𝑆)
3114oneli 5873 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 ∈ On)
321inteqi 4511 . . . . . . . . . . . . . . . . 17 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
3332eleq2i 2722 . . . . . . . . . . . . . . . 16 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
34 oveq2 6698 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑧))
3534sseq2d 3666 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
3635onnminsb 7046 . . . . . . . . . . . . . . . 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 7660 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 𝑧) ∈ On)
395, 38mpan 706 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ On → (𝐴 +𝑜 𝑧) ∈ On)
40 ontri1 5795 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ (𝐴 +𝑜 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
414, 39, 40sylancr 696 . . . . . . . . . . . . . . . 16 (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
4241con2bid 343 . . . . . . . . . . . . . . 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 5870 . . . . . . . . . . . . . 14 Ord 𝐵
46 ordsucss 7060 . . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5026, 49eqsstrd 3672 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
5150rexlimiva 3057 . . . . . . . . 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 7657 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴 +𝑜 𝑆) = 𝑧 𝑆(𝐴 +𝑜 𝑧))
545, 53mpan 706 . . . . . . . . . 10 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +𝑜 𝑆) = 𝑧 𝑆(𝐴 +𝑜 𝑧))
55 iunss 4593 . . . . . . . . . . 11 ( 𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵)
564onelssi 5874 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝑧) ∈ 𝐵 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5744, 56syl 17 . . . . . . . . . . 11 (𝑧 𝑆 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5855, 57mprgbir 2956 . . . . . . . . . 10 𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵
5954, 58syl6eqss 3688 . . . . . . . . 9 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
6059a1d 25 . . . . . . . 8 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
6122, 52, 603jaoi 1431 . . . . . . 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 3340 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦))
644, 7, 63mp2an 708 . . . . . . . 8 𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦)
65 nfcv 2793 . . . . . . . . . 10 𝑦𝐵
66 nfcv 2793 . . . . . . . . . . 11 𝑦𝐴
67 nfcv 2793 . . . . . . . . . . 11 𝑦 +𝑜
68 nfrab1 3152 . . . . . . . . . . . 12 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
6968nfint 4518 . . . . . . . . . . 11 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
7066, 67, 69nfov 6716 . . . . . . . . . 10 𝑦(𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7165, 70nfss 3629 . . . . . . . . 9 𝑦 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
72 oveq2 6698 . . . . . . . . . 10 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
7372sseq2d 3666 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
7471, 73onminsb 7041 . . . . . . . 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 6701 . . . . . . 7 (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7775, 76sseqtr4i 3671 . . . . . 6 𝐵 ⊆ (𝐴 +𝑜 𝑆)
7862, 77jctir 560 . . . . 5 (𝐴𝐵 → ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +𝑜 𝑆)))
79 eqss 3651 . . . . 5 ((𝐴 +𝑜 𝑆) = 𝐵 ↔ ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +𝑜 𝑆)))
8078, 79sylibr 224 . . . 4 (𝐴𝐵 → (𝐴 +𝑜 𝑆) = 𝐵)
81 oveq2 6698 . . . . . 6 (𝑥 = 𝑆 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑆))
8281eqeq1d 2653 . . . . 5 (𝑥 = 𝑆 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑆) = 𝐵))
8382rspcev 3340 . . . 4 (( 𝑆 ∈ On ∧ (𝐴 +𝑜 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
8414, 80, 83sylancr 696 . . 3 (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
85 eqtr3 2672 . . . . 5 (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
86 oacan 7673 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦) ↔ 𝑥 = 𝑦))
875, 86mp3an1 1451 . . . . 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 3006 . . 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 6698 . . . 4 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
9291eqeq1d 2653 . . 3 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑦) = 𝐵))
9392reu4 3433 . 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 383   ∨ w3o 1053   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  ∃!wreu 2943  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ∩ cint 4507  ∪ ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690   +𝑜 coa 7602 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609 This theorem is referenced by:  oawordeu  7680
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