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Theorem altopthsn 34933
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 34930 . . 3 𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2 df-altop 34930 . . 3 𝐶, 𝐷⟫ = {{𝐶}, {𝐶, {𝐷}}}
31, 2eqeq12i 2751 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4 snex 5432 . . . . . 6 {𝐴} ∈ V
5 prex 5433 . . . . . 6 {𝐴, {𝐵}} ∈ V
6 snex 5432 . . . . . 6 {𝐶} ∈ V
7 prex 5433 . . . . . 6 {𝐶, {𝐷}} ∈ V
84, 5, 6, 7preq12b 4852 . . . . 5 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})))
9 simpl 484 . . . . . 6 (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) → {𝐴} = {𝐶})
10 snsspr1 4818 . . . . . . . . 9 {𝐴} ⊆ {𝐴, {𝐵}}
11 sseq2 4009 . . . . . . . . 9 ({𝐴, {𝐵}} = {𝐶} → ({𝐴} ⊆ {𝐴, {𝐵}} ↔ {𝐴} ⊆ {𝐶}))
1210, 11mpbii 232 . . . . . . . 8 ({𝐴, {𝐵}} = {𝐶} → {𝐴} ⊆ {𝐶})
1312adantl 483 . . . . . . 7 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} ⊆ {𝐶})
14 snsspr1 4818 . . . . . . . . 9 {𝐶} ⊆ {𝐶, {𝐷}}
15 sseq2 4009 . . . . . . . . 9 ({𝐴} = {𝐶, {𝐷}} → ({𝐶} ⊆ {𝐴} ↔ {𝐶} ⊆ {𝐶, {𝐷}}))
1614, 15mpbiri 258 . . . . . . . 8 ({𝐴} = {𝐶, {𝐷}} → {𝐶} ⊆ {𝐴})
1716adantr 482 . . . . . . 7 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐶} ⊆ {𝐴})
1813, 17eqssd 4000 . . . . . 6 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} = {𝐶})
199, 18jaoi 856 . . . . 5 ((({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})) → {𝐴} = {𝐶})
208, 19sylbi 216 . . . 4 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐴} = {𝐶})
21 uneq1 4157 . . . . . . . . . 10 ({𝐴} = {𝐶} → ({𝐴} ∪ {{𝐵}}) = ({𝐶} ∪ {{𝐵}}))
22 df-pr 4632 . . . . . . . . . 10 {𝐴, {𝐵}} = ({𝐴} ∪ {{𝐵}})
23 df-pr 4632 . . . . . . . . . 10 {𝐶, {𝐵}} = ({𝐶} ∪ {{𝐵}})
2421, 22, 233eqtr4g 2798 . . . . . . . . 9 ({𝐴} = {𝐶} → {𝐴, {𝐵}} = {𝐶, {𝐵}})
2524preq2d 4745 . . . . . . . 8 ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐴}, {𝐶, {𝐵}}})
26 preq1 4738 . . . . . . . 8 ({𝐴} = {𝐶} → {{𝐴}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
2725, 26eqtrd 2773 . . . . . . 7 ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
2827eqeq1d 2735 . . . . . 6 ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
2928biimpd 228 . . . . 5 ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
30 prex 5433 . . . . . . 7 {𝐶, {𝐵}} ∈ V
3130, 7preqr2 4851 . . . . . 6 ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
32 snex 5432 . . . . . . 7 {𝐵} ∈ V
33 snex 5432 . . . . . . 7 {𝐷} ∈ V
3432, 33preqr2 4851 . . . . . 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 518 . . 3 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
38 preq2 4739 . . . . 5 ({𝐵} = {𝐷} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
3938preq2d 4745 . . . 4 ({𝐵} = {𝐷} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4027, 39sylan9eq 2793 . . 3 (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4137, 40impbii 208 . 2 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
423, 41bitri 275 1 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wo 846   = wceq 1542  cun 3947  wss 3949  {csn 4629  {cpr 4631  caltop 34928
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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-altop 34930
This theorem is referenced by:  altopeq12  34934  altopth1  34937  altopth2  34938  altopthg  34939  altopthbg  34940
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