Theorem bj-2upln1upl 37042

Description: A couple is never equal to a monuple. It is in order to have this "non-clashing" result that tagging was used. Without tagging, we would have ⦅𝐴, ∅⦆ = ⦅𝐴⦆. Note that in the context of Morse tuples, it is natural to define the 0-tuple as the empty set. Therefore, the present theorem together with bj-1upln0 37027 and bj-2upln0 37041 tell us that an m-tuple may equal an n-tuple only when m = n, at least for m, n <= 2, but this result would extend as soon as we define n-tuples for higher values of n. (Contributed by BJ, 21-Apr-2019.)

Assertion

Ref	Expression
bj-2upln1upl	⊢ ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆

Proof of Theorem bj-2upln1upl

Step	Hyp	Ref	Expression
1		xpundi 5723	. . . . . . 7 ⊢ ({∅} × (tag 𝐴 ∪ tag 𝐶)) = (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
2	1	difeq2i 4098	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
3		incom 4184	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4		xp01disjl 8504	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = ∅
5	3, 4	eqtr3i 2760	. . . . . . . 8 ⊢ (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
6		disjdif2 4455	. . . . . . . 8 ⊢ ((({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅ → (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ({1o} × tag 𝐵))
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ({1o} × tag 𝐵)
8		1oex 8490	. . . . . . . . . 10 ⊢ 1o ∈ V
9	8	snnz 4752	. . . . . . . . 9 ⊢ {1o} ≠ ∅
10		bj-tagn0 36997	. . . . . . . . 9 ⊢ tag 𝐵 ≠ ∅
11	9, 10	pm3.2i 470	. . . . . . . 8 ⊢ ({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
12		xpnz 6148	. . . . . . . 8 ⊢ (({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1o} × tag 𝐵) ≠ ∅)
13	11, 12	mpbi 230	. . . . . . 7 ⊢ ({1o} × tag 𝐵) ≠ ∅
14	7, 13	eqnetri 3002	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
15	2, 14	eqnetrri 3003	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
16		0pss 4422	. . . . 5 ⊢ (∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ↔ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅)
17	15, 16	mpbir 231	. . . 4 ⊢ ∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
18		ssun2 4154	. . . . . . . 8 ⊢ ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
19		sscon 4118	. . . . . . . 8 ⊢ (({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)) → (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶)))
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶))
21		ssun2 4154	. . . . . . . 8 ⊢ ({1o} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
22		ssdif 4119	. . . . . . . 8 ⊢ (({1o} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) → (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶)) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶)))
23	21, 22	ax-mp 5	. . . . . . 7 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶)) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
24	20, 23	sstri 3968	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
25		df-bj-2upl 37029	. . . . . . . 8 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
26		df-bj-1upl 37016	. . . . . . . . 9 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
27	26	uneq1i 4139	. . . . . . . 8 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
28	25, 27	eqtri 2758	. . . . . . 7 ⊢ ⦅𝐴, 𝐵⦆ = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
29	28	difeq1i 4097	. . . . . 6 ⊢ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶)) = ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
30	24, 29	sseqtrri 4008	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
31		df-bj-1upl 37016	. . . . . 6 ⊢ ⦅𝐶⦆ = ({∅} × tag 𝐶)
32	31	difeq2i 4098	. . . . 5 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
33	30, 32	sseqtrri 4008	. . . 4 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
34		psssstr 4084	. . . 4 ⊢ ((∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ∧ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)) → ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆))
35	17, 33, 34	mp2an 692	. . 3 ⊢ ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36		0pss 4422	. . 3 ⊢ (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
37	35, 36	mpbi 230	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
38		difn0 4342	. 2 ⊢ ((⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅ → ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆)
39	37, 38	ax-mp 5	1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆

Syntax hints:   ∧ wa 395   = wceq 1540   ≠ wne 2932   ∖ cdif 3923   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926   ⊊ wpss 3927  ∅c0 4308  {csn 4601   × cxp 5652  1_oc1o 8473  tag bj-ctag 36992  ⦅bj-c1upl 37015  ⦅bj-c2uple 37028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-suc 6358  df-1o 8480  df-bj-tag 36993  df-bj-1upl 37016  df-bj-2upl 37029

This theorem is referenced by: (None)