Theorem altopthsn 33417

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

Step	Hyp	Ref	Expression
1		df-altop 33414	. . 3 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2		df-altop 33414	. . 3 ⊢ ⟪𝐶, 𝐷⟫ = {{𝐶}, {𝐶, {𝐷}}}
3	1, 2	eqeq12i 2836	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4		snex 5323	. . . . . 6 ⊢ {𝐴} ∈ V
5		prex 5324	. . . . . 6 ⊢ {𝐴, {𝐵}} ∈ V
6		snex 5323	. . . . . 6 ⊢ {𝐶} ∈ V
7		prex 5324	. . . . . 6 ⊢ {𝐶, {𝐷}} ∈ V
8	4, 5, 6, 7	preq12b 4774	. . . . 5 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})))
9		simpl 485	. . . . . 6 ⊢ (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) → {𝐴} = {𝐶})
10		snsspr1 4740	. . . . . . . . 9 ⊢ {𝐴} ⊆ {𝐴, {𝐵}}
11		sseq2 3992	. . . . . . . . 9 ⊢ ({𝐴, {𝐵}} = {𝐶} → ({𝐴} ⊆ {𝐴, {𝐵}} ↔ {𝐴} ⊆ {𝐶}))
12	10, 11	mpbii 235	. . . . . . . 8 ⊢ ({𝐴, {𝐵}} = {𝐶} → {𝐴} ⊆ {𝐶})
13	12	adantl 484	. . . . . . 7 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} ⊆ {𝐶})
14		snsspr1 4740	. . . . . . . . 9 ⊢ {𝐶} ⊆ {𝐶, {𝐷}}
15		sseq2 3992	. . . . . . . . 9 ⊢ ({𝐴} = {𝐶, {𝐷}} → ({𝐶} ⊆ {𝐴} ↔ {𝐶} ⊆ {𝐶, {𝐷}}))
16	14, 15	mpbiri 260	. . . . . . . 8 ⊢ ({𝐴} = {𝐶, {𝐷}} → {𝐶} ⊆ {𝐴})
17	16	adantr 483	. . . . . . 7 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐶} ⊆ {𝐴})
18	13, 17	eqssd 3983	. . . . . 6 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} = {𝐶})
19	9, 18	jaoi 853	. . . . 5 ⊢ ((({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})) → {𝐴} = {𝐶})
20	8, 19	sylbi 219	. . . 4 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐴} = {𝐶})
21		uneq1 4131	. . . . . . . . . 10 ⊢ ({𝐴} = {𝐶} → ({𝐴} ∪ {{𝐵}}) = ({𝐶} ∪ {{𝐵}}))
22		df-pr 4563	. . . . . . . . . 10 ⊢ {𝐴, {𝐵}} = ({𝐴} ∪ {{𝐵}})
23		df-pr 4563	. . . . . . . . . 10 ⊢ {𝐶, {𝐵}} = ({𝐶} ∪ {{𝐵}})
24	21, 22, 23	3eqtr4g 2881	. . . . . . . . 9 ⊢ ({𝐴} = {𝐶} → {𝐴, {𝐵}} = {𝐶, {𝐵}})
25	24	preq2d 4669	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐴}, {𝐶, {𝐵}}})
26		preq1 4662	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
27	25, 26	eqtrd 2856	. . . . . . 7 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
28	27	eqeq1d 2823	. . . . . 6 ⊢ ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
29	28	biimpd 231	. . . . 5 ⊢ ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
30		prex 5324	. . . . . . 7 ⊢ {𝐶, {𝐵}} ∈ V
31	30, 7	preqr2 4773	. . . . . 6 ⊢ ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
32		snex 5323	. . . . . . 7 ⊢ {𝐵} ∈ V
33		snex 5323	. . . . . . 7 ⊢ {𝐷} ∈ V
34	32, 33	preqr2 4773	. . . . . 6 ⊢ ({𝐶, {𝐵}} = {𝐶, {𝐷}} → {𝐵} = {𝐷})
35	31, 34	syl 17	. . . . 5 ⊢ ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐵} = {𝐷})
36	29, 35	syl6com 37	. . . 4 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → ({𝐴} = {𝐶} → {𝐵} = {𝐷}))
37	20, 36	jcai 519	. . 3 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
38		preq2 4663	. . . . 5 ⊢ ({𝐵} = {𝐷} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
39	38	preq2d 4669	. . . 4 ⊢ ({𝐵} = {𝐷} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
40	27, 39	sylan9eq 2876	. . 3 ⊢ (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
41	37, 40	impbii 211	. 2 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
42	3, 41	bitri 277	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∪ cun 3933   ⊆ wss 3935  {csn 4560  {cpr 4562  ⟪caltop 33412

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4561  df-pr 4563  df-altop 33414

This theorem is referenced by:  altopeq12  33418  altopth1  33421  altopth2  33422  altopthg  33423  altopthbg  33424