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Theorem bj-2upln1upl 32694
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 32679 and bj-2upln0 32693 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 5137 . . . . . . 7 ({∅} × (tag 𝐴 ∪ tag 𝐶)) = (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
21difeq2i 3708 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
3 incom 3788 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = (({1𝑜} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4 bj-disjsn01 32619 . . . . . . . . . 10 ({∅} ∩ {1𝑜}) = ∅
5 xpdisj1 5519 . . . . . . . . . 10 (({∅} ∩ {1𝑜}) = ∅ → (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = ∅)
64, 5ax-mp 5 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1𝑜} × tag 𝐵)) = ∅
73, 6eqtr3i 2645 . . . . . . . 8 (({1𝑜} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
8 disjdif2 4024 . . . . . . . 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 32620 . . . . . . . . . 10 1𝑜 ∈ V
1110snnz 4284 . . . . . . . . 9 {1𝑜} ≠ ∅
12 bj-tagn0 32649 . . . . . . . . 9 tag 𝐵 ≠ ∅
1311, 12pm3.2i 471 . . . . . . . 8 ({1𝑜} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
14 xpnz 5517 . . . . . . . 8 (({1𝑜} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1𝑜} × tag 𝐵) ≠ ∅)
1513, 14mpbi 220 . . . . . . 7 ({1𝑜} × tag 𝐵) ≠ ∅
169, 15eqnetri 2860 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
172, 16eqnetrri 2861 . . . . 5 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
18 0pss 3990 . . . . 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 3760 . . . . . . . 8 ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
21 sscon 3727 . . . . . . . 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 3760 . . . . . . . 8 ({1𝑜} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
24 ssdif 3728 . . . . . . . 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 3596 . . . . . 6 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
27 df-bj-2upl 32681 . . . . . . . 8 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
28 df-bj-1upl 32668 . . . . . . . . 9 𝐴⦆ = ({∅} × tag 𝐴)
2928uneq1i 3746 . . . . . . . 8 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
3027, 29eqtri 2643 . . . . . . 7 𝐴, 𝐵⦆ = (({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵))
3130difeq1i 3707 . . . . . 6 (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶)) = ((({∅} × tag 𝐴) ∪ ({1𝑜} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
3226, 31sseqtr4i 3622 . . . . 5 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
33 df-bj-1upl 32668 . . . . . 6 𝐶⦆ = ({∅} × tag 𝐶)
3433difeq2i 3708 . . . . 5 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
3532, 34sseqtr4i 3622 . . . 4 (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36 psssstr 3696 . . . 4 ((∅ ⊊ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ∧ (({1𝑜} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)) → ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆))
3719, 35, 36mp2an 707 . . 3 ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
38 0pss 3990 . . 3 (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
3937, 38mpbi 220 . 2 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
40 difn0 3922 . 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 1480  wne 2790  cdif 3556  cun 3557  cin 3558  wss 3559  wpss 3560  c0 3896  {csn 4153   × cxp 5077  1𝑜c1o 7505  tag bj-ctag 32644  bj-c1upl 32667  bj-c2uple 32680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-suc 5693  df-1o 7512  df-bj-tag 32645  df-bj-1upl 32668  df-bj-2upl 32681
This theorem is referenced by: (None)
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