Theorem altopthsn 36204

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 36201	. . 3 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2		df-altop 36201	. . 3 ⊢ ⟪𝐶, 𝐷⟫ = {{𝐶}, {𝐶, {𝐷}}}
3	1, 2	eqeq12i 2759	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4		snex 5371	. . . . . 6 ⊢ {𝐴} ∈ V
5		prex 5370	. . . . . 6 ⊢ {𝐴, {𝐵}} ∈ V
6		snex 5371	. . . . . 6 ⊢ {𝐶} ∈ V
7		prex 5370	. . . . . 6 ⊢ {𝐶, {𝐷}} ∈ V
8	4, 5, 6, 7	preq12b 4784	. . . . 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 4748	. . . . . . . . 9 ⊢ {𝐴} ⊆ {𝐴, {𝐵}}
11		sseq2 3943	. . . . . . . . 9 ⊢ ({𝐴, {𝐵}} = {𝐶} → ({𝐴} ⊆ {𝐴, {𝐵}} ↔ {𝐴} ⊆ {𝐶}))
12	10, 11	mpbii 235	. . . . . . . 8 ⊢ ({𝐴, {𝐵}} = {𝐶} → {𝐴} ⊆ {𝐶})
13	12	adantl 483	. . . . . . 7 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} ⊆ {𝐶})
14		snsspr1 4748	. . . . . . . . 9 ⊢ {𝐶} ⊆ {𝐶, {𝐷}}
15		sseq2 3943	. . . . . . . . 9 ⊢ ({𝐴} = {𝐶, {𝐷}} → ({𝐶} ⊆ {𝐴} ↔ {𝐶} ⊆ {𝐶, {𝐷}}))
16	14, 15	mpbiri 260	. . . . . . . 8 ⊢ ({𝐴} = {𝐶, {𝐷}} → {𝐶} ⊆ {𝐴})
17	16	adantr 482	. . . . . . 7 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐶} ⊆ {𝐴})
18	13, 17	eqssd 3934	. . . . . 6 ⊢ (({𝐴} = {𝐶, {𝐷}} ∧ {𝐴, {𝐵}} = {𝐶}) → {𝐴} = {𝐶})
19	9, 18	jaoi 864	. . . . 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 4094	. . . . . . . . . 10 ⊢ ({𝐴} = {𝐶} → ({𝐴} ∪ {{𝐵}}) = ({𝐶} ∪ {{𝐵}}))
22		df-pr 4561	. . . . . . . . . 10 ⊢ {𝐴, {𝐵}} = ({𝐴} ∪ {{𝐵}})
23		df-pr 4561	. . . . . . . . . 10 ⊢ {𝐶, {𝐵}} = ({𝐶} ∪ {{𝐵}})
24	21, 22, 23	3eqtr4g 2801	. . . . . . . . 9 ⊢ ({𝐴} = {𝐶} → {𝐴, {𝐵}} = {𝐶, {𝐵}})
25	24	preq2d 4675	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐴}, {𝐶, {𝐵}}})
26		preq1 4668	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
27	25, 26	eqtrd 2776	. . . . . . 7 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
28	27	eqeq1d 2743	. . . . . 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 5370	. . . . . . 7 ⊢ {𝐶, {𝐵}} ∈ V
31	30, 7	preqr2 4783	. . . . . 6 ⊢ ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
32		snex 5371	. . . . . . 7 ⊢ {𝐵} ∈ V
33		snex 5371	. . . . . . 7 ⊢ {𝐷} ∈ V
34	32, 33	preqr2 4783	. . . . . 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 522	. . 3 ⊢ ({{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
38		preq2 4669	. . . . 5 ⊢ ({𝐵} = {𝐷} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
39	38	preq2d 4675	. . . 4 ⊢ ({𝐵} = {𝐷} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
40	27, 39	sylan9eq 2796	. . 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 397   ∨ wo 854   = wceq 1548   ∪ cun 3883   ⊆ wss 3885  {csn 4558  {cpr 4560  ⟪caltop 36199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-un 3890  df-ss 3902  df-sn 4559  df-pr 4561  df-altop 36201

This theorem is referenced by:  altopeq12  36205  altopth1  36208  altopth2  36209  altopthg  36210  altopthbg  36211