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Theorem bj-2upln1upl 33137
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 33122 and bj-2upln0 33136 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
StepHypRef Expression
1 xpundi 5205 . . . . . . 7 ({∅} × (tag 𝐴 ∪ tag 𝐶)) = (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
21difeq2i 3758 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
3 incom 3838 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = (({1𝑜} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4 bj-disjsn01 33062 . . . . . . . . . 10 ({∅} ∩ {1𝑜}) = ∅
5 xpdisj1 5590 . . . . . . . . . 10 (({∅} ∩ {1𝑜}) = ∅ → (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = ∅)
64, 5ax-mp 5 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = ∅
73, 6eqtr3i 2675 . . . . . . . 8 (({1𝑜} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
8 disjdif2 4080 . . . . . . . 8 ((({1𝑜} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅ → (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ({1𝑜} × tag 𝐵))
97, 8ax-mp 5 . . . . . . 7 (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ({1𝑜} × tag 𝐵)
10 bj-1ex 33063 . . . . . . . . . 10 1𝑜 ∈ V
1110snnz 4340 . . . . . . . . 9 {1𝑜} ≠ ∅
12 bj-tagn0 33092 . . . . . . . . 9 tag 𝐵 ≠ ∅
1311, 12pm3.2i 470 . . . . . . . 8 ({1𝑜} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
14 xpnz 5588 . . . . . . . 8 (({1𝑜} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1𝑜} × tag 𝐵) ≠ ∅)
1513, 14mpbi 220 . . . . . . 7 ({1𝑜} × tag 𝐵) ≠ ∅
169, 15eqnetri 2893 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
172, 16eqnetrri 2894 . . . . 5 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
18 0pss 4046 . . . . 5 (∅ ⊊ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ↔ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅)
1917, 18mpbir 221 . . . 4 ∅ ⊊ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
20 ssun2 3810 . . . . . . . 8 ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
21 sscon 3777 . . . . . . . 8 (({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)) → (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (({1𝑜} × tag 𝐵) ∖ ({∅} × tag 𝐶)))
2220, 21ax-mp 5 . . . . . . 7 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (({1𝑜} × tag 𝐵) ∖ ({∅} × tag 𝐶))
23 ssun2 3810 . . . . . . . 8 ({1𝑜} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
24 ssdif 3778 . . . . . . . 8 (({1𝑜} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) → (({1𝑜} × tag 𝐵) ∖ ({∅} × tag 𝐶)) ⊆ ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶)))
2523, 24ax-mp 5 . . . . . . 7 (({1𝑜} × tag 𝐵) ∖ ({∅} × tag 𝐶)) ⊆ ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
2622, 25sstri 3645 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
27 df-bj-2upl 33124 . . . . . . . 8 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
28 df-bj-1upl 33111 . . . . . . . . 9 𝐴⦆ = ({∅} × tag 𝐴)
2928uneq1i 3796 . . . . . . . 8 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
3027, 29eqtri 2673 . . . . . . 7 𝐴, 𝐵⦆ = (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
3130difeq1i 3757 . . . . . 6 (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶)) = ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
3226, 31sseqtr4i 3671 . . . . 5 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
33 df-bj-1upl 33111 . . . . . 6 𝐶⦆ = ({∅} × tag 𝐶)
3433difeq2i 3758 . . . . 5 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
3532, 34sseqtr4i 3671 . . . 4 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36 psssstr 3746 . . . 4 ((∅ ⊊ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ∧ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)) → ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆))
3719, 35, 36mp2an 708 . . 3 ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
38 0pss 4046 . . 3 (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
3937, 38mpbi 220 . 2 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
40 difn0 3976 . 2 ((⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅ → ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆)
4139, 40ax-mp 5 1 𝐴, 𝐵⦆ ≠ ⦅𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wne 2823  cdif 3604  cun 3605  cin 3606  wss 3607  wpss 3608  c0 3948  {csn 4210   × cxp 5141  1𝑜c1o 7598  tag bj-ctag 33087  bj-c1upl 33110  bj-c2uple 33123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-suc 5767  df-1o 7605  df-bj-tag 33088  df-bj-1upl 33111  df-bj-2upl 33124
This theorem is referenced by: (None)
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