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Theorem altopthsn 35943
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 35940 . . 3 𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2 df-altop 35940 . . 3 𝐶, 𝐷⟫ = {{𝐶}, {𝐶, {𝐷}}}
31, 2eqeq12i 2753 . 2 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4 snex 5442 . . . . . 6 {𝐴} ∈ V
5 prex 5443 . . . . . 6 {𝐴, {𝐵}} ∈ V
6 snex 5442 . . . . . 6 {𝐶} ∈ V
7 prex 5443 . . . . . 6 {𝐶, {𝐷}} ∈ V
84, 5, 6, 7preq12b 4855 . . . . 5 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})))
9 simpl 482 . . . . . 6 (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) → {𝐴} = {𝐶})
10 snsspr1 4819 . . . . . . . . 9 {𝐴} ⊆ {𝐴, {𝐵}}
11 sseq2 4022 . . . . . . . . 9 ({𝐴, {𝐵}} = {𝐶} → ({𝐴} ⊆ {𝐴, {𝐵}} ↔ {𝐴} ⊆ {𝐶}))
1210, 11mpbii 233 . . . . . . . 8 ({𝐴, {𝐵}} = {𝐶} → {𝐴} ⊆ {𝐶})
1312adantl 481 . . . . . . 7 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} ⊆ {𝐶})
14 snsspr1 4819 . . . . . . . . 9 {𝐶} ⊆ {𝐶, {𝐷}}
15 sseq2 4022 . . . . . . . . 9 ({𝐴} = {𝐶, {𝐷}} → ({𝐶} ⊆ {𝐴} ↔ {𝐶} ⊆ {𝐶, {𝐷}}))
1614, 15mpbiri 258 . . . . . . . 8 ({𝐴} = {𝐶, {𝐷}} → {𝐶} ⊆ {𝐴})
1716adantr 480 . . . . . . 7 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐶} ⊆ {𝐴})
1813, 17eqssd 4013 . . . . . 6 (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} = {𝐶})
199, 18jaoi 857 . . . . 5 ((({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})) → {𝐴} = {𝐶})
208, 19sylbi 217 . . . 4 ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐴} = {𝐶})
21 uneq1 4171 . . . . . . . . . 10 ({𝐴} = {𝐶} → ({𝐴} ∪ {{𝐵}}) = ({𝐶} ∪ {{𝐵}}))
22 df-pr 4634 . . . . . . . . . 10 {𝐴, {𝐵}} = ({𝐴} ∪ {{𝐵}})
23 df-pr 4634 . . . . . . . . . 10 {𝐶, {𝐵}} = ({𝐶} ∪ {{𝐵}})
2421, 22, 233eqtr4g 2800 . . . . . . . . 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 2775 . . . . . . 7 ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
2827eqeq1d 2737 . . . . . 6 ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
2928biimpd 229 . . . . 5 ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
30 prex 5443 . . . . . . 7 {𝐶, {𝐵}} ∈ V
3130, 7preqr2 4854 . . . . . 6 ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
32 snex 5442 . . . . . . 7 {𝐵} ∈ V
33 snex 5442 . . . . . . 7 {𝐷} ∈ V
3432, 33preqr2 4854 . . . . . 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 516 . . 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 2795 . . 3 (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4137, 40impbii 209 . 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 206  wa 395  wo 847   = wceq 1537  cun 3961  wss 3963  {csn 4631  {cpr 4633  caltop 35938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-altop 35940
This theorem is referenced by:  altopeq12  35944  altopth1  35947  altopth2  35948  altopthg  35949  altopthbg  35950
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