Theorem bj-2upln1upl 34338

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 34323 and bj-2upln0 34337 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 5622	. . . . . . 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 4180	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4		xp01disjl 8123	. . . . . . . . 9 ⊢ (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = ∅
5	3, 4	eqtr3i 2848	. . . . . . . 8 ⊢ (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
6		disjdif2 4430	. . . . . . . 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 8112	. . . . . . . . . 10 ⊢ 1o ∈ V
9	8	snnz 4713	. . . . . . . . 9 ⊢ {1o} ≠ ∅
10		bj-tagn0 34293	. . . . . . . . 9 ⊢ tag 𝐵 ≠ ∅
11	9, 10	pm3.2i 473	. . . . . . . 8 ⊢ ({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
12		xpnz 6018	. . . . . . . 8 ⊢ (({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1o} × tag 𝐵) ≠ ∅)
13	11, 12	mpbi 232	. . . . . . 7 ⊢ ({1o} × tag 𝐵) ≠ ∅
14	7, 13	eqnetri 3088	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
15	2, 14	eqnetrri 3089	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
16		0pss 4398	. . . . 5 ⊢ (∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ↔ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅)
17	15, 16	mpbir 233	. . . 4 ⊢ ∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
18		ssun2 4151	. . . . . . . 8 ⊢ ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
19		sscon 4117	. . . . . . . 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 4151	. . . . . . . 8 ⊢ ({1o} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
22		ssdif 4118	. . . . . . . 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 3978	. . . . . 6 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
25		df-bj-2upl 34325	. . . . . . . 8 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
26		df-bj-1upl 34312	. . . . . . . . 9 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
27	26	uneq1i 4137	. . . . . . . 8 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
28	25, 27	eqtri 2846	. . . . . . 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 4006	. . . . 5 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
31		df-bj-1upl 34312	. . . . . 6 ⊢ ⦅𝐶⦆ = ({∅} × tag 𝐶)
32	31	difeq2i 4098	. . . . 5 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
33	30, 32	sseqtrri 4006	. . . 4 ⊢ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
34		psssstr 4085	. . . 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 690	. . 3 ⊢ ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36		0pss 4398	. . 3 ⊢ (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
37	35, 36	mpbi 232	. 2 ⊢ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
38		difn0 4326	. 2 ⊢ ((⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅ → ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆)
39	37, 38	ax-mp 5	1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆

Syntax hints:   ∧ wa 398   = wceq 1537   ≠ wne 3018   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938   ⊊ wpss 3939  ∅c0 4293  {csn 4569   × cxp 5555  1_oc1o 8097  tag bj-ctag 34288  ⦅bj-c1upl 34311  ⦅bj-c2uple 34324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-ord 6196  df-on 6197  df-suc 6199  df-1o 8104  df-bj-tag 34289  df-bj-1upl 34312  df-bj-2upl 34325

This theorem is referenced by: (None)