Theorem altopthsn 33415

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 33412	. . 3 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2		df-altop 33412	. . 3 ⊢ ⟪𝐶, 𝐷⟫ = {{𝐶}, {𝐶, {𝐷}}}
3	1, 2	eqeq12i 2834	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
4		snex 5322	. . . . . 6 ⊢ {𝐴} ∈ V
5		prex 5323	. . . . . 6 ⊢ {𝐴, {𝐵}} ∈ V
6		snex 5322	. . . . . 6 ⊢ {𝐶} ∈ V
7		prex 5323	. . . . . 6 ⊢ {𝐶, {𝐷}} ∈ V
8	4, 5, 6, 7	preq12b 4773	. . . . 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 4739	. . . . . . . . 9 ⊢ {𝐴} ⊆ {𝐴, {𝐵}}
11		sseq2 3991	. . . . . . . . 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 4739	. . . . . . . . 9 ⊢ {𝐶} ⊆ {𝐶, {𝐷}}
15		sseq2 3991	. . . . . . . . 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 3982	. . . . . 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 4130	. . . . . . . . . 10 ⊢ ({𝐴} = {𝐶} → ({𝐴} ∪ {{𝐵}}) = ({𝐶} ∪ {{𝐵}}))
22		df-pr 4562	. . . . . . . . . 10 ⊢ {𝐴, {𝐵}} = ({𝐴} ∪ {{𝐵}})
23		df-pr 4562	. . . . . . . . . 10 ⊢ {𝐶, {𝐵}} = ({𝐶} ∪ {{𝐵}})
24	21, 22, 23	3eqtr4g 2879	. . . . . . . . 9 ⊢ ({𝐴} = {𝐶} → {𝐴, {𝐵}} = {𝐶, {𝐵}})
25	24	preq2d 4668	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐴}, {𝐶, {𝐵}}})
26		preq1 4661	. . . . . . . 8 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
27	25, 26	eqtrd 2854	. . . . . . 7 ⊢ ({𝐴} = {𝐶} → {{𝐴}, {𝐴, {𝐵}}} = {{𝐶}, {𝐶, {𝐵}}})
28	27	eqeq1d 2821	. . . . . 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 5323	. . . . . . 7 ⊢ {𝐶, {𝐵}} ∈ V
31	30, 7	preqr2 4772	. . . . . 6 ⊢ ({{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
32		snex 5322	. . . . . . 7 ⊢ {𝐵} ∈ V
33		snex 5322	. . . . . . 7 ⊢ {𝐷} ∈ V
34	32, 33	preqr2 4772	. . . . . 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 4662	. . . . 5 ⊢ ({𝐵} = {𝐷} → {𝐶, {𝐵}} = {𝐶, {𝐷}})
39	38	preq2d 4668	. . . 4 ⊢ ({𝐵} = {𝐷} → {{𝐶}, {𝐶, {𝐵}}} = {{𝐶}, {𝐶, {𝐷}}})
40	27, 39	sylan9eq 2874	. . 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 1531   ∪ cun 3932   ⊆ wss 3934  {csn 4559  {cpr 4561  ⟪caltop 33410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-pr 4562  df-altop 33412

This theorem is referenced by:  altopeq12  33416  altopth1  33419  altopth2  33420  altopthg  33421  altopthbg  33422