Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  altopthsn Structured version   Visualization version   GIF version

Theorem altopthsn 32395
Description: Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
altopthsn (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))

Proof of Theorem altopthsn
StepHypRef Expression
1 df-altop 32392 . . 3 𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2 df-altop 32392 . . 3 𝐶, 𝐷⟫ = {{𝐶}, {𝐶, {𝐷}}}
31, 2eqeq12i 2774 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4 snex 5057 . . . . . 6 {𝐴} ∈ V
5 prex 5058 . . . . . 6 {𝐴, {𝐵}} ∈ V
6 snex 5057 . . . . . 6 {𝐶} ∈ V
7 prex 5058 . . . . . 6 {𝐶, {𝐷}} ∈ V
84, 5, 6, 7preq12b 4526 . . . . 5 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})))
9 simpl 474 . . . . . 6 (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) → {𝐴} = {𝐶})
10 snsspr1 4490 . . . . . . . . 9 {𝐴} ⊆ {𝐴, {𝐵}}
11 sseq2 3768 . . . . . . . . 9 ({𝐴, {𝐵}} = {𝐶} → ({𝐴} ⊆ {𝐴, {𝐵}} ↔ {𝐴} ⊆ {𝐶}))
1210, 11mpbii 223 . . . . . . . 8 ({𝐴, {𝐵}} = {𝐶} → {𝐴} ⊆ {𝐶})
1312adantl 473 . . . . . . 7 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} ⊆ {𝐶})
14 snsspr1 4490 . . . . . . . . 9 {𝐶} ⊆ {𝐶, {𝐷}}
15 sseq2 3768 . . . . . . . . 9 ({𝐴} = {𝐶, {𝐷}} → ({𝐶} ⊆ {𝐴} ↔ {𝐶} ⊆ {𝐶, {𝐷}}))
1614, 15mpbiri 248 . . . . . . . 8 ({𝐴} = {𝐶, {𝐷}} → {𝐶} ⊆ {𝐴})
1716adantr 472 . . . . . . 7 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐶} ⊆ {𝐴})
1813, 17eqssd 3761 . . . . . 6 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} = {𝐶})
199, 18jaoi 393 . . . . 5 ((({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})) → {𝐴} = {𝐶})
208, 19sylbi 207 . . . 4 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐴} = {𝐶})
21 uneq1 3903 . . . . . . . . . 10 ({𝐴} = {𝐶} → ({𝐴} ∪ {{𝐵}}) = ({𝐶} ∪ {{𝐵}}))
22 df-pr 4324 . . . . . . . . . 10 {𝐴, {𝐵}} = ({𝐴} ∪ {{𝐵}})
23 df-pr 4324 . . . . . . . . . 10 {𝐶, {𝐵}} = ({𝐶} ∪ {{𝐵}})
2421, 22, 233eqtr4g 2819 . . . . . . . . 9 ({𝐴} = {𝐶} → {𝐴, {𝐵}} = {𝐶, {𝐵}})
2524preq2d 4419 . . . . . . . 8 ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐴}, {𝐶, {𝐵}}})
26 preq1 4412 . . . . . . . 8 ({𝐴} = {𝐶} → {{𝐴}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
2725, 26eqtrd 2794 . . . . . . 7 ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
2827eqeq1d 2762 . . . . . 6 ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
2928biimpd 219 . . . . 5 ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
30 prex 5058 . . . . . . 7 {𝐶, {𝐵}} ∈ V
3130, 7preqr2 4525 . . . . . 6 ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
32 snex 5057 . . . . . . 7 {𝐵} ∈ V
33 snex 5057 . . . . . . 7 {𝐷} ∈ V
3432, 33preqr2 4525 . . . . . 6 ({𝐶, {𝐵}} = {𝐶, {𝐷}} → {𝐵} = {𝐷})
3531, 34syl 17 . . . . 5 ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐵} = {𝐷})
3629, 35syl6com 37 . . . 4 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → ({𝐴} = {𝐶} → {𝐵} = {𝐷}))
3720, 36jcai 560 . . 3 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
38 preq2 4413 . . . . 5 ({𝐵} = {𝐷} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
3938preq2d 4419 . . . 4 ({𝐵} = {𝐷} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4027, 39sylan9eq 2814 . . 3 (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4137, 40impbii 199 . 2 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
423, 41bitri 264 1 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383   = wceq 1632  cun 3713  wss 3715  {csn 4321  {cpr 4323  caltop 32390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324  df-altop 32392
This theorem is referenced by:  altopeq12  32396  altopth1  32399  altopth2  32400  altopthg  32401  altopthbg  32402
  Copyright terms: Public domain W3C validator