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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 34220 | . 2 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-1upln0 34218 | . . . . 5 ⊢ ⦅𝐴⦆ ≠ ∅ | |
3 | 0pss 4392 | . . . . 5 ⊢ (∅ ⊊ ⦅𝐴⦆ ↔ ⦅𝐴⦆ ≠ ∅) | |
4 | 2, 3 | mpbir 232 | . . . 4 ⊢ ∅ ⊊ ⦅𝐴⦆ |
5 | ssun1 4145 | . . . 4 ⊢ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
6 | psssstr 4080 | . . . 4 ⊢ ((∅ ⊊ ⦅𝐴⦆ ∧ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) → ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) | |
7 | 4, 5, 6 | mp2an 688 | . . 3 ⊢ ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) |
8 | 0pss 4392 | . . 3 ⊢ (∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅) | |
9 | 7, 8 | mpbi 231 | . 2 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅ |
10 | 1, 9 | eqnetri 3083 | 1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 3013 ∪ cun 3931 ⊆ wss 3933 ⊊ wpss 3934 ∅c0 4288 {csn 4557 × cxp 5546 1oc1o 8084 tag bj-ctag 34183 ⦅bj-c1upl 34206 ⦅bj-c2uple 34219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-bj-tag 34184 df-bj-1upl 34207 df-bj-2upl 34220 |
This theorem is referenced by: (None) |
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