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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 36490 | . 2 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-1upln0 36488 | . . . . 5 ⊢ ⦅𝐴⦆ ≠ ∅ | |
3 | 0pss 4445 | . . . . 5 ⊢ (∅ ⊊ ⦅𝐴⦆ ↔ ⦅𝐴⦆ ≠ ∅) | |
4 | 2, 3 | mpbir 230 | . . . 4 ⊢ ∅ ⊊ ⦅𝐴⦆ |
5 | ssun1 4172 | . . . 4 ⊢ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
6 | psssstr 4104 | . . . 4 ⊢ ((∅ ⊊ ⦅𝐴⦆ ∧ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) → ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) | |
7 | 4, 5, 6 | mp2an 691 | . . 3 ⊢ ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) |
8 | 0pss 4445 | . . 3 ⊢ (∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅) | |
9 | 7, 8 | mpbi 229 | . 2 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅ |
10 | 1, 9 | eqnetri 3008 | 1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2937 ∪ cun 3945 ⊆ wss 3947 ⊊ wpss 3948 ∅c0 4323 {csn 4629 × cxp 5676 1oc1o 8479 tag bj-ctag 36453 ⦅bj-c1upl 36476 ⦅bj-c2uple 36489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-bj-tag 36454 df-bj-1upl 36477 df-bj-2upl 36490 |
This theorem is referenced by: (None) |
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