Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-2upln1upl Structured version   Visualization version   GIF version

Theorem bj-2upln1upl 33340
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 33325 and bj-2upln0 33339 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 5384 . . . . . . 7 ({∅} × (tag 𝐴 ∪ tag 𝐶)) = (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
21difeq2i 3935 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
3 incom 4015 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = (({1𝑜} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4 bj-disjsn01 33266 . . . . . . . . . 10 ({∅} ∩ {1𝑜}) = ∅
5 xpdisj1 5779 . . . . . . . . . 10 (({∅} ∩ {1𝑜}) = ∅ → (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = ∅)
64, 5ax-mp 5 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = ∅
73, 6eqtr3i 2841 . . . . . . . 8 (({1𝑜} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
8 disjdif2 4254 . . . . . . . 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 1oex 7813 . . . . . . . . . 10 1𝑜 ∈ V
1110snnz 4510 . . . . . . . . 9 {1𝑜} ≠ ∅
12 bj-tagn0 33295 . . . . . . . . 9 tag 𝐵 ≠ ∅
1311, 12pm3.2i 458 . . . . . . . 8 ({1𝑜} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
14 xpnz 5777 . . . . . . . 8 (({1𝑜} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1𝑜} × tag 𝐵) ≠ ∅)
1513, 14mpbi 221 . . . . . . 7 ({1𝑜} × tag 𝐵) ≠ ∅
169, 15eqnetri 3059 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
172, 16eqnetrri 3060 . . . . 5 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
18 0pss 4222 . . . . 5 (∅ ⊊ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ↔ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅)
1917, 18mpbir 222 . . . 4 ∅ ⊊ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
20 ssun2 3987 . . . . . . . 8 ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
21 sscon 3954 . . . . . . . 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 3987 . . . . . . . 8 ({1𝑜} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
24 ssdif 3955 . . . . . . . 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 3818 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
27 df-bj-2upl 33327 . . . . . . . 8 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
28 df-bj-1upl 33314 . . . . . . . . 9 𝐴⦆ = ({∅} × tag 𝐴)
2928uneq1i 3973 . . . . . . . 8 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
3027, 29eqtri 2839 . . . . . . 7 𝐴, 𝐵⦆ = (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
3130difeq1i 3934 . . . . . 6 (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶)) = ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
3226, 31sseqtr4i 3846 . . . . 5 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
33 df-bj-1upl 33314 . . . . . 6 𝐶⦆ = ({∅} × tag 𝐶)
3433difeq2i 3935 . . . . 5 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
3532, 34sseqtr4i 3846 . . . 4 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36 psssstr 3922 . . . 4 ((∅ ⊊ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ∧ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)) → ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆))
3719, 35, 36mp2an 675 . . 3 ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
38 0pss 4222 . . 3 (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
3937, 38mpbi 221 . 2 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
40 difn0 4155 . 2 ((⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅ → ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆)
4139, 40ax-mp 5 1 𝐴, 𝐵⦆ ≠ ⦅𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wne 2989  cdif 3777  cun 3778  cin 3779  wss 3780  wpss 3781  c0 4127  {csn 4381   × cxp 5322  1𝑜c1o 7798  tag bj-ctag 33290  bj-c1upl 33313  bj-c2uple 33326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-tr 4958  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-ord 5952  df-on 5953  df-suc 5955  df-1o 7805  df-bj-tag 33291  df-bj-1upl 33314  df-bj-2upl 33327
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator