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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 34326 | . 2 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-1upln0 34324 | . . . . 5 ⊢ ⦅𝐴⦆ ≠ ∅ | |
3 | 0pss 4396 | . . . . 5 ⊢ (∅ ⊊ ⦅𝐴⦆ ↔ ⦅𝐴⦆ ≠ ∅) | |
4 | 2, 3 | mpbir 233 | . . . 4 ⊢ ∅ ⊊ ⦅𝐴⦆ |
5 | ssun1 4148 | . . . 4 ⊢ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
6 | psssstr 4083 | . . . 4 ⊢ ((∅ ⊊ ⦅𝐴⦆ ∧ ⦅𝐴⦆ ⊆ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) → ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) | |
7 | 4, 5, 6 | mp2an 690 | . . 3 ⊢ ∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) |
8 | 0pss 4396 | . . 3 ⊢ (∅ ⊊ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅) | |
9 | 7, 8 | mpbi 232 | . 2 ⊢ (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) ≠ ∅ |
10 | 1, 9 | eqnetri 3086 | 1 ⊢ ⦅𝐴, 𝐵⦆ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 3016 ∪ cun 3934 ⊆ wss 3936 ⊊ wpss 3937 ∅c0 4291 {csn 4567 × cxp 5553 1oc1o 8095 tag bj-ctag 34289 ⦅bj-c1upl 34312 ⦅bj-c2uple 34325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-bj-tag 34290 df-bj-1upl 34313 df-bj-2upl 34326 |
This theorem is referenced by: (None) |
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