MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oawordeulem Structured version   Visualization version   GIF version

Theorem oawordeulem 7839
Description: Lemma for oawordex 7842. (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 3847 . . . . . 6 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
31, 2eqsstri 3795 . . . . 5 𝑆 ⊆ On
4 oawordeulem.2 . . . . . . 7 𝐵 ∈ On
5 oawordeulem.1 . . . . . . . 8 𝐴 ∈ On
6 oaword2 7838 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
74, 5, 6mp2an 683 . . . . . . 7 𝐵 ⊆ (𝐴 +𝑜 𝐵)
8 oveq2 6850 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
98sseq2d 3793 . . . . . . . 8 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
109, 1elrab2 3523 . . . . . . 7 (𝐵𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
114, 7, 10mpbir2an 702 . . . . . 6 𝐵𝑆
1211ne0ii 4088 . . . . 5 𝑆 ≠ ∅
13 oninton 7198 . . . . 5 ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
143, 12, 13mp2an 683 . . . 4 𝑆 ∈ On
15 onzsl 7244 . . . . . . . 8 ( 𝑆 ∈ On ↔ ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆)))
1614, 15mpbi 221 . . . . . . 7 ( 𝑆 = ∅ ∨ ∃𝑧 ∈ On 𝑆 = suc 𝑧 ∨ ( 𝑆 ∈ V ∧ Lim 𝑆))
17 oveq2 6850 . . . . . . . . . . 11 ( 𝑆 = ∅ → (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 ∅))
18 oa0 7801 . . . . . . . . . . . 12 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
195, 18ax-mp 5 . . . . . . . . . . 11 (𝐴 +𝑜 ∅) = 𝐴
2017, 19syl6eq 2815 . . . . . . . . . 10 ( 𝑆 = ∅ → (𝐴 +𝑜 𝑆) = 𝐴)
2120sseq1d 3792 . . . . . . . . 9 ( 𝑆 = ∅ → ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐴𝐵))
2221biimprd 239 . . . . . . . 8 ( 𝑆 = ∅ → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
23 oveq2 6850 . . . . . . . . . . . 12 ( 𝑆 = suc 𝑧 → (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 suc 𝑧))
24 oasuc 7809 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
255, 24mpan 681 . . . . . . . . . . . 12 (𝑧 ∈ On → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
2623, 25sylan9eqr 2821 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +𝑜 𝑆) = suc (𝐴 +𝑜 𝑧))
27 vex 3353 . . . . . . . . . . . . . . 15 𝑧 ∈ V
2827sucid 5987 . . . . . . . . . . . . . 14 𝑧 ∈ suc 𝑧
29 eleq2 2833 . . . . . . . . . . . . . 14 ( 𝑆 = suc 𝑧 → (𝑧 𝑆𝑧 ∈ suc 𝑧))
3028, 29mpbiri 249 . . . . . . . . . . . . 13 ( 𝑆 = suc 𝑧𝑧 𝑆)
3114oneli 6015 . . . . . . . . . . . . . 14 (𝑧 𝑆𝑧 ∈ On)
321inteqi 4637 . . . . . . . . . . . . . . . . 17 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
3332eleq2i 2836 . . . . . . . . . . . . . . . 16 (𝑧 𝑆𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
34 oveq2 6850 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑧))
3534sseq2d 3793 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
3635onnminsb 7202 . . . . . . . . . . . . . . . 16 (𝑧 ∈ On → (𝑧 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
3733, 36syl5bi 233 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝑧 𝑆 → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
38 oacl 7820 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 𝑧) ∈ On)
395, 38mpan 681 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ On → (𝐴 +𝑜 𝑧) ∈ On)
40 ontri1 5942 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ (𝐴 +𝑜 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
414, 39, 40sylancr 581 . . . . . . . . . . . . . . . 16 (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
4241con2bid 345 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → ((𝐴 +𝑜 𝑧) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
4337, 42sylibrd 250 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑧 𝑆 → (𝐴 +𝑜 𝑧) ∈ 𝐵))
4431, 43mpcom 38 . . . . . . . . . . . . 13 (𝑧 𝑆 → (𝐴 +𝑜 𝑧) ∈ 𝐵)
454onordi 6012 . . . . . . . . . . . . . 14 Ord 𝐵
46 ordsucss 7216 . . . . . . . . . . . . . 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 473 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5026, 49eqsstrd 3799 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑆 = suc 𝑧) → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
5150rexlimiva 3175 . . . . . . . . 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 7817 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ( 𝑆 ∈ V ∧ Lim 𝑆)) → (𝐴 +𝑜 𝑆) = 𝑧 𝑆(𝐴 +𝑜 𝑧))
545, 53mpan 681 . . . . . . . . . 10 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +𝑜 𝑆) = 𝑧 𝑆(𝐴 +𝑜 𝑧))
55 iunss 4717 . . . . . . . . . . 11 ( 𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵 ↔ ∀𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵)
564onelssi 6016 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝑧) ∈ 𝐵 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5744, 56syl 17 . . . . . . . . . . 11 (𝑧 𝑆 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
5855, 57mprgbir 3074 . . . . . . . . . 10 𝑧 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵
5954, 58syl6eqss 3815 . . . . . . . . 9 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴 +𝑜 𝑆) ⊆ 𝐵)
6059a1d 25 . . . . . . . 8 (( 𝑆 ∈ V ∧ Lim 𝑆) → (𝐴𝐵 → (𝐴 +𝑜 𝑆) ⊆ 𝐵))
6122, 52, 603jaoi 1552 . . . . . . 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 3461 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦))
644, 7, 63mp2an 683 . . . . . . . 8 𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦)
65 nfcv 2907 . . . . . . . . . 10 𝑦𝐵
66 nfcv 2907 . . . . . . . . . . 11 𝑦𝐴
67 nfcv 2907 . . . . . . . . . . 11 𝑦 +𝑜
68 nfrab1 3270 . . . . . . . . . . . 12 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
6968nfint 4643 . . . . . . . . . . 11 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
7066, 67, 69nfov 6872 . . . . . . . . . 10 𝑦(𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7165, 70nfss 3754 . . . . . . . . 9 𝑦 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
72 oveq2 6850 . . . . . . . . . 10 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
7372sseq2d 3793 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
7471, 73onminsb 7197 . . . . . . . 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 6853 . . . . . . 7 (𝐴 +𝑜 𝑆) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7775, 76sseqtr4i 3798 . . . . . 6 𝐵 ⊆ (𝐴 +𝑜 𝑆)
7862, 77jctir 516 . . . . 5 (𝐴𝐵 → ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +𝑜 𝑆)))
79 eqss 3776 . . . . 5 ((𝐴 +𝑜 𝑆) = 𝐵 ↔ ((𝐴 +𝑜 𝑆) ⊆ 𝐵𝐵 ⊆ (𝐴 +𝑜 𝑆)))
8078, 79sylibr 225 . . . 4 (𝐴𝐵 → (𝐴 +𝑜 𝑆) = 𝐵)
81 oveq2 6850 . . . . . 6 (𝑥 = 𝑆 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑆))
8281eqeq1d 2767 . . . . 5 (𝑥 = 𝑆 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑆) = 𝐵))
8382rspcev 3461 . . . 4 (( 𝑆 ∈ On ∧ (𝐴 +𝑜 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
8414, 80, 83sylancr 581 . . 3 (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
85 eqtr3 2786 . . . . 5 (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
86 oacan 7833 . . . . . 6 ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦) ↔ 𝑥 = 𝑦))
875, 86mp3an1 1572 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦) ↔ 𝑥 = 𝑦))
8885, 87syl5ib 235 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦))
8988rgen2a 3124 . . 3 𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦)
9084, 89jctir 516 . 2 (𝐴𝐵 → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦)))
91 oveq2 6850 . . . 4 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
9291eqeq1d 2767 . . 3 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑦) = 𝐵))
9392reu4 3559 . 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 225 1 (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3o 1106   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  ∃!wreu 3057  {crab 3059  Vcvv 3350  wss 3732  c0 4079   cint 4633   ciun 4676  Ord word 5907  Oncon0 5908  Lim wlim 5909  suc csuc 5910  (class class class)co 6842   +𝑜 coa 7761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768
This theorem is referenced by:  oawordeu  7840
  Copyright terms: Public domain W3C validator