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Theorem bj-2upln1upl 34404
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 34389 and bj-2upln0 34403 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 5607 . . . . . . 7 ({∅} × (tag 𝐴 ∪ tag 𝐶)) = (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
21difeq2i 4082 . . . . . 6 (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
3 incom 4163 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶)))
4 xp01disjl 8117 . . . . . . . . 9 (({∅} × (tag 𝐴 ∪ tag 𝐶)) ∩ ({1o} × tag 𝐵)) = ∅
53, 4eqtr3i 2849 . . . . . . . 8 (({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅
6 disjdif2 4411 . . . . . . . 8 ((({1o} × tag 𝐵) ∩ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ∅ → (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ({1o} × tag 𝐵))
75, 6ax-mp 5 . . . . . . 7 (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) = ({1o} × tag 𝐵)
8 1oex 8106 . . . . . . . . . 10 1o ∈ V
98snnz 4696 . . . . . . . . 9 {1o} ≠ ∅
10 bj-tagn0 34359 . . . . . . . . 9 tag 𝐵 ≠ ∅
119, 10pm3.2i 474 . . . . . . . 8 ({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅)
12 xpnz 6003 . . . . . . . 8 (({1o} ≠ ∅ ∧ tag 𝐵 ≠ ∅) ↔ ({1o} × tag 𝐵) ≠ ∅)
1311, 12mpbi 233 . . . . . . 7 ({1o} × tag 𝐵) ≠ ∅
147, 13eqnetri 3084 . . . . . 6 (({1o} × tag 𝐵) ∖ ({∅} × (tag 𝐴 ∪ tag 𝐶))) ≠ ∅
152, 14eqnetrri 3085 . . . . 5 (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅
16 0pss 4379 . . . . 5 (∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ↔ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ≠ ∅)
1715, 16mpbir 234 . . . 4 ∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)))
18 ssun2 4135 . . . . . . . 8 ({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))
19 sscon 4101 . . . . . . . 8 (({∅} × tag 𝐶) ⊆ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶)) → (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶)))
2018, 19ax-mp 5 . . . . . . 7 (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶))
21 ssun2 4135 . . . . . . . 8 ({1o} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
22 ssdif 4102 . . . . . . . 8 (({1o} × tag 𝐵) ⊆ (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) → (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶)) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶)))
2321, 22ax-mp 5 . . . . . . 7 (({1o} × tag 𝐵) ∖ ({∅} × tag 𝐶)) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
2420, 23sstri 3962 . . . . . 6 (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
25 df-bj-2upl 34391 . . . . . . . 8 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
26 df-bj-1upl 34378 . . . . . . . . 9 𝐴⦆ = ({∅} × tag 𝐴)
2726uneq1i 4121 . . . . . . . 8 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
2825, 27eqtri 2847 . . . . . . 7 𝐴, 𝐵⦆ = (({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵))
2928difeq1i 4081 . . . . . 6 (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶)) = ((({∅} × tag 𝐴) ∪ ({1o} × tag 𝐵)) ∖ ({∅} × tag 𝐶))
3024, 29sseqtrri 3990 . . . . 5 (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
31 df-bj-1upl 34378 . . . . . 6 𝐶⦆ = ({∅} × tag 𝐶)
3231difeq2i 4082 . . . . 5 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) = (⦅𝐴, 𝐵⦆ ∖ ({∅} × tag 𝐶))
3330, 32sseqtrri 3990 . . . 4 (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
34 psssstr 4069 . . . 4 ((∅ ⊊ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ∧ (({1o} × tag 𝐵) ∖ (({∅} × tag 𝐴) ∪ ({∅} × tag 𝐶))) ⊆ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)) → ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆))
3517, 33, 34mp2an 691 . . 3 ∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆)
36 0pss 4379 . . 3 (∅ ⊊ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ↔ (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅)
3735, 36mpbi 233 . 2 (⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅
38 difn0 4307 . 2 ((⦅𝐴, 𝐵⦆ ∖ ⦅𝐶⦆) ≠ ∅ → ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆)
3937, 38ax-mp 5 1 𝐴, 𝐵⦆ ≠ ⦅𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wne 3014  cdif 3916  cun 3917  cin 3918  wss 3919  wpss 3920  c0 4276  {csn 4550   × cxp 5540  1oc1o 8091  tag bj-ctag 34354  bj-c1upl 34377  bj-c2uple 34390
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-ord 6181  df-on 6182  df-suc 6184  df-1o 8098  df-bj-tag 34355  df-bj-1upl 34378  df-bj-2upl 34391
This theorem is referenced by: (None)
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