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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 37013 | . 2 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
| 2 | bj-1upln0 37011 | . . . . 5 ⊢ ⦅𝐴⦆ ≠ ∅ | |
| 3 | 0pss 4446 | . . . . 5 ⊢ (∅ ⊊ ⦅𝐴⦆ ↔ ⦅𝐴⦆ ≠ ∅) | |
| 4 | 2, 3 | mpbir 231 | . . . 4 ⊢ ∅ ⊊ ⦅𝐴⦆ | 
| 5 | ssun1 4177 | . . . 4 ⊢ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
| 6 | psssstr 4108 | . . . 4 ⊢ ((∅ ⊊ ⦅𝐴⦆ ∧ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) → ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) | |
| 7 | 4, 5, 6 | mp2an 692 | . . 3 ⊢ ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | 
| 8 | 0pss 4446 | . . 3 ⊢ (∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅) | |
| 9 | 7, 8 | mpbi 230 | . 2 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅ | 
| 10 | 1, 9 | eqnetri 3010 | 1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ≠ wne 2939 ∪ cun 3948 ⊆ wss 3950 ⊊ wpss 3951 ∅c0 4332 {csn 4625 × cxp 5682 1oc1o 8500 tag bj-ctag 36976 ⦅bj-c1upl 36999 ⦅bj-c2uple 37012 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-bj-tag 36977 df-bj-1upl 37000 df-bj-2upl 37013 | 
| This theorem is referenced by: (None) | 
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