Theorem altopthsn 33535

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 33532	. . 3 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2		df-altop 33532	. . 3 ⊢ ⟪𝐶, 𝐷⟫ = {{𝐶}, {𝐶, {𝐷}}}
3	1, 2	eqeq12i 2813	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4		snex 5297	. . . . . 6 ⊢ {𝐴} ∈ V
5		prex 5298	. . . . . 6 ⊢ {𝐴, {𝐵}} ∈ V
6		snex 5297	. . . . . 6 ⊢ {𝐶} ∈ V
7		prex 5298	. . . . . 6 ⊢ {𝐶, {𝐷}} ∈ V
8	4, 5, 6, 7	preq12b 4741	. . . . 5 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})))
9		simpl 486	. . . . . 6 ⊢ (({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) → {𝐴} = {𝐶})
10		snsspr1 4707	. . . . . . . . 9 ⊢ {𝐴} ⊆ {𝐴, {𝐵}}
11		sseq2 3941	. . . . . . . . 9 ⊢ ({𝐴, {𝐵}} = {𝐶} → ({𝐴} ⊆ {𝐴, {𝐵}} ↔ {𝐴} ⊆ {𝐶}))
12	10, 11	mpbii 236	. . . . . . . 8 ⊢ ({𝐴, {𝐵}} = {𝐶} → {𝐴} ⊆ {𝐶})
13	12	adantl 485	. . . . . . 7 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} ⊆ {𝐶})
14		snsspr1 4707	. . . . . . . . 9 ⊢ {𝐶} ⊆ {𝐶, {𝐷}}
15		sseq2 3941	. . . . . . . . 9 ⊢ ({𝐴} = {𝐶, {𝐷}} → ({𝐶} ⊆ {𝐴} ↔ {𝐶} ⊆ {𝐶, {𝐷}}))
16	14, 15	mpbiri 261	. . . . . . . 8 ⊢ ({𝐴} = {𝐶, {𝐷}} → {𝐶} ⊆ {𝐴})
17	16	adantr 484	. . . . . . 7 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐶} ⊆ {𝐴})
18	13, 17	eqssd 3932	. . . . . 6 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} = {𝐶})
19	9, 18	jaoi 854	. . . . 5 ⊢ ((({𝐴} = {𝐶} ∧ {𝐴, {𝐵}} = {𝐶, {𝐷}}) ∨ ({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶})) → {𝐴} = {𝐶})
20	8, 19	sylbi 220	. . . 4 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐴} = {𝐶})
21		uneq1 4083	. . . . . . . . . 10 ⊢ ({𝐴} = {𝐶} → ({𝐴} ∪ {{𝐵}}) = ({𝐶} ∪ {{𝐵}}))
22		df-pr 4528	. . . . . . . . . 10 ⊢ {𝐴, {𝐵}} = ({𝐴} ∪ {{𝐵}})
23		df-pr 4528	. . . . . . . . . 10 ⊢ {𝐶, {𝐵}} = ({𝐶} ∪ {{𝐵}})
24	21, 22, 23	3eqtr4g 2858	. . . . . . . . 9 ⊢ ({𝐴} = {𝐶} → {𝐴, {𝐵}} = {𝐶, {𝐵}})
25	24	preq2d 4636	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐴}, {𝐶, {𝐵}}})
26		preq1 4629	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
27	25, 26	eqtrd 2833	. . . . . . 7 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
28	27	eqeq1d 2800	. . . . . 6 ⊢ ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
29	28	biimpd 232	. . . . 5 ⊢ ({𝐴} = {𝐶} → ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}}))
30		prex 5298	. . . . . . 7 ⊢ {𝐶, {𝐵}} ∈ V
31	30, 7	preqr2 4740	. . . . . 6 ⊢ ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
32		snex 5297	. . . . . . 7 ⊢ {𝐵} ∈ V
33		snex 5297	. . . . . . 7 ⊢ {𝐷} ∈ V
34	32, 33	preqr2 4740	. . . . . 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 520	. . 3 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
38		preq2 4630	. . . . 5 ⊢ ({𝐵} = {𝐷} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
39	38	preq2d 4636	. . . 4 ⊢ ({𝐵} = {𝐷} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
40	27, 39	sylan9eq 2853	. . 3 ⊢ (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
41	37, 40	impbii 212	. 2 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
42	3, 41	bitri 278	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∪ cun 3879   ⊆ wss 3881  {csn 4525  {cpr 4527  ⟪caltop 33530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-altop 33532

This theorem is referenced by:  altopeq12  33536  altopth1  33539  altopth2  33540  altopthg  33541  altopthbg  33542