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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 33941 | . 2 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-1upln0 33939 | . . . . 5 ⊢ ⦅𝐴⦆ ≠ ∅ | |
3 | 0pss 4312 | . . . . 5 ⊢ (∅ ⊊ ⦅𝐴⦆ ↔ ⦅𝐴⦆ ≠ ∅) | |
4 | 2, 3 | mpbir 232 | . . . 4 ⊢ ∅ ⊊ ⦅𝐴⦆ |
5 | ssun1 4071 | . . . 4 ⊢ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
6 | psssstr 4006 | . . . 4 ⊢ ((∅ ⊊ ⦅𝐴⦆ ∧ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) → ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) | |
7 | 4, 5, 6 | mp2an 688 | . . 3 ⊢ ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) |
8 | 0pss 4312 | . . 3 ⊢ (∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅) | |
9 | 7, 8 | mpbi 231 | . 2 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅ |
10 | 1, 9 | eqnetri 3053 | 1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2983 ∪ cun 3859 ⊆ wss 3861 ⊊ wpss 3862 ∅c0 4213 {csn 4474 × cxp 5444 1oc1o 7949 tag bj-ctag 33904 ⦅bj-c1upl 33927 ⦅bj-c2uple 33940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-br 4965 df-opab 5027 df-xp 5452 df-rel 5453 df-cnv 5454 df-bj-tag 33905 df-bj-1upl 33928 df-bj-2upl 33941 |
This theorem is referenced by: (None) |
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