Theorem bj-2upln1upl 38012

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⦆. 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 37997 and bj-2upln0 38011 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 5719	. . . . . . 7 ⊢ ({∅} × (tag 𝐴 ∪ tag 𝐶)) = (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
2	1	difeq2i 4060	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
3		incom 4106	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4		xp01disjl 8501	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = ∅
5	3, 4	eqtr3i 2785	. . . . . . . 8 ⊢ (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
6		disjdif2 4435	. . . . . . . 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 8487	. . . . . . . . . 10 ⊢ 1o ∈ V
9	8	snnz 4736	. . . . . . . . 9 ⊢ {1o} ≠ ∅
10		bj-tagn0 37967	. . . . . . . . 9 ⊢ tag 𝐵 ≠ ∅
11	9, 10	pm3.2i 476	. . . . . . . 8 ⊢ ({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
12		xpnz 6149	. . . . . . . 8 ⊢ (({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1o} × tag 𝐵) ≠ ∅)
13	11, 12	mpbi 233	. . . . . . 7 ⊢ ({1o} × tag 𝐵) ≠ ∅
14	7, 13	eqnetri 3025	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
15	2, 14	eqnetrri 3026	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
16		0pss 4359	. . . . 5 ⊢ (∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ↔ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅)
17	15, 16	mpbir 234	. . . 4 ⊢ ∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
18		ssun2 4177	. . . . . . . 8 ⊢ ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
19		sscon 4079	. . . . . . . 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 4177	. . . . . . . 8 ⊢ ({1o} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
22		ssdif 4080	. . . . . . . 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 3917	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
25		df-bj-2upl 37999	. . . . . . . 8 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
26		df-bj-1upl 37986	. . . . . . . . 9 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
27	26	uneq1i 4163	. . . . . . . 8 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
28	25, 27	eqtri 2783	. . . . . . 7 ⊢ ⦅𝐴, 𝐵⦆ = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
29	28	difeq1i 4059	. . . . . 6 ⊢ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶)) = ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
30	24, 29	sseqtrri 3957	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
31		df-bj-1upl 37986	. . . . . 6 ⊢ ⦅𝐶⦆ = ({∅} × tag 𝐶)
32	31	difeq2i 4060	. . . . 5 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
33	30, 32	sseqtrri 3957	. . . 4 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
34		psssstr 4037	. . . 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 705	. . 3 ⊢ ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36		0pss 4359	. . 3 ⊢ (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
37	35, 36	mpbi 233	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
38		difn0 4314	. 2 ⊢ ((⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅ → ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆)
39	37, 38	ax-mp 5	1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ≠ wne 2955   ⊆ wss 3895   ⊊ wpss 4013   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4583   × cxp 5648  1_oc1o 8470  tag bj-ctag 37962  ⦅bj-c1upl 37985  ⦅bj-c2uple 37998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-xp 5656  df-rel 5657  df-suc 6368  df-1o 8477  df-bj-tag 37963  df-bj-1upl 37986  df-bj-2upl 37999

This theorem is used by: (None)