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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-2upln0 | Structured version Visualization version GIF version |
Description: A couple is nonempty. (Contributed by BJ, 21-Apr-2019.) |
Ref | Expression |
---|---|
bj-2upln0 | ⊢ ⦅𝐴, 𝐵⦆ ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-2upl 35589 | . 2 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-1upln0 35587 | . . . . 5 ⊢ ⦅𝐴⦆ ≠ ∅ | |
3 | 0pss 4424 | . . . . 5 ⊢ (∅ ⊊ ⦅𝐴⦆ ↔ ⦅𝐴⦆ ≠ ∅) | |
4 | 2, 3 | mpbir 230 | . . . 4 ⊢ ∅ ⊊ ⦅𝐴⦆ |
5 | ssun1 4152 | . . . 4 ⊢ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
6 | psssstr 4086 | . . . 4 ⊢ ((∅ ⊊ ⦅𝐴⦆ ∧ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) → ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) | |
7 | 4, 5, 6 | mp2an 690 | . . 3 ⊢ ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) |
8 | 0pss 4424 | . . 3 ⊢ (∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅) | |
9 | 7, 8 | mpbi 229 | . 2 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅ |
10 | 1, 9 | eqnetri 3010 | 1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2939 ∪ cun 3926 ⊆ wss 3928 ⊊ wpss 3929 ∅c0 4302 {csn 4606 × cxp 5651 1oc1o 8425 tag bj-ctag 35552 ⦅bj-c1upl 35575 ⦅bj-c2uple 35588 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pr 5404 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-br 5126 df-opab 5188 df-xp 5659 df-rel 5660 df-cnv 5661 df-bj-tag 35553 df-bj-1upl 35576 df-bj-2upl 35589 |
This theorem is referenced by: (None) |
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