Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  intubeu Structured version   Visualization version   GIF version

Theorem intubeu 47609
Description: Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.)
Assertion
Ref Expression
intubeu (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) ↔ 𝐶 = {𝑥𝐵𝐴𝑥}))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem intubeu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssint 4969 . . . . . . 7 (𝐶 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐶𝑦)
2 sseq2 4009 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32ralrab 3690 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐶𝑦 ↔ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
41, 3bitri 275 . . . . . 6 (𝐶 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
54biimpri 227 . . . . 5 (∀𝑦𝐵 (𝐴𝑦𝐶𝑦) → 𝐶 {𝑥𝐵𝐴𝑥})
65adantl 483 . . . 4 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 {𝑥𝐵𝐴𝑥})
7 sseq2 4009 . . . . . . 7 (𝑧 = 𝐶 → (𝐴𝑧𝐴𝐶))
8 simpll 766 . . . . . . 7 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶𝐵)
9 simplr 768 . . . . . . 7 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐴𝐶)
107, 8, 9elrabd 3686 . . . . . 6 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 ∈ {𝑧𝐵𝐴𝑧})
11 sseq2 4009 . . . . . . 7 (𝑧 = 𝑥 → (𝐴𝑧𝐴𝑥))
1211cbvrabv 3443 . . . . . 6 {𝑧𝐵𝐴𝑧} = {𝑥𝐵𝐴𝑥}
1310, 12eleqtrdi 2844 . . . . 5 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 ∈ {𝑥𝐵𝐴𝑥})
14 intss1 4968 . . . . 5 (𝐶 ∈ {𝑥𝐵𝐴𝑥} → {𝑥𝐵𝐴𝑥} ⊆ 𝐶)
1513, 14syl 17 . . . 4 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → {𝑥𝐵𝐴𝑥} ⊆ 𝐶)
166, 15eqssd 4000 . . 3 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 = {𝑥𝐵𝐴𝑥})
1716expl 459 . 2 (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 = {𝑥𝐵𝐴𝑥}))
18 ssintub 4971 . . . 4 𝐴 {𝑥𝐵𝐴𝑥}
19 sseq2 4009 . . . 4 (𝐶 = {𝑥𝐵𝐴𝑥} → (𝐴𝐶𝐴 {𝑥𝐵𝐴𝑥}))
2018, 19mpbiri 258 . . 3 (𝐶 = {𝑥𝐵𝐴𝑥} → 𝐴𝐶)
21 eqimss 4041 . . . 4 (𝐶 = {𝑥𝐵𝐴𝑥} → 𝐶 {𝑥𝐵𝐴𝑥})
2221, 4sylib 217 . . 3 (𝐶 = {𝑥𝐵𝐴𝑥} → ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
2320, 22jca 513 . 2 (𝐶 = {𝑥𝐵𝐴𝑥} → (𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)))
2417, 23impbid1 224 1 (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) ↔ 𝐶 = {𝑥𝐵𝐴𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  {crab 3433  wss 3949   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-int 4952
This theorem is referenced by:  ipolubdm  47612  ipolub  47613
  Copyright terms: Public domain W3C validator