Theorem bj-2upln1upl 37200

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 37185 and bj-2upln0 37199 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 5694	. . . . . . 7 ⊢ ({∅} × (tag 𝐴 ∪ tag 𝐶)) = (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
2	1	difeq2i 4076	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
3		incom 4162	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4		xp01disjl 8421	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = ∅
5	3, 4	eqtr3i 2762	. . . . . . . 8 ⊢ (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
6		disjdif2 4433	. . . . . . . 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 8409	. . . . . . . . . 10 ⊢ 1o ∈ V
9	8	snnz 4734	. . . . . . . . 9 ⊢ {1o} ≠ ∅
10		bj-tagn0 37155	. . . . . . . . 9 ⊢ tag 𝐵 ≠ ∅
11	9, 10	pm3.2i 470	. . . . . . . 8 ⊢ ({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
12		xpnz 6118	. . . . . . . 8 ⊢ (({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1o} × tag 𝐵) ≠ ∅)
13	11, 12	mpbi 230	. . . . . . 7 ⊢ ({1o} × tag 𝐵) ≠ ∅
14	7, 13	eqnetri 3003	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
15	2, 14	eqnetrri 3004	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
16		0pss 4400	. . . . 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 4132	. . . . . . . 8 ⊢ ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
19		sscon 4096	. . . . . . . 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 4132	. . . . . . . 8 ⊢ ({1o} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
22		ssdif 4097	. . . . . . . 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 3944	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
25		df-bj-2upl 37187	. . . . . . . 8 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
26		df-bj-1upl 37174	. . . . . . . . 9 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
27	26	uneq1i 4117	. . . . . . . 8 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
28	25, 27	eqtri 2760	. . . . . . 7 ⊢ ⦅𝐴, 𝐵⦆ = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
29	28	difeq1i 4075	. . . . . 6 ⊢ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶)) = ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
30	24, 29	sseqtrri 3984	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
31		df-bj-1upl 37174	. . . . . 6 ⊢ ⦅𝐶⦆ = ({∅} × tag 𝐶)
32	31	difeq2i 4076	. . . . 5 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
33	30, 32	sseqtrri 3984	. . . 4 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
34		psssstr 4062	. . . 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 693	. . 3 ⊢ ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36		0pss 4400	. . 3 ⊢ (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
37	35, 36	mpbi 230	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
38		difn0 4320	. 2 ⊢ ((⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅ → ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆)
39	37, 38	ax-mp 5	1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆

Syntax hints:   ∧ wa 395   = wceq 1542   ≠ wne 2933   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4286  {csn 4581   × cxp 5623  1_oc1o 8392  tag bj-ctag 37150  ⦅bj-c1upl 37173  ⦅bj-c2uple 37186

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-xp 5631  df-rel 5632  df-suc 6324  df-1o 8399  df-bj-tag 37151  df-bj-1upl 37174  df-bj-2upl 37187

This theorem is referenced by: (None)