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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdeq0 | GIF version |
Description: Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
bdeq0 | ⊢ BOUNDED 𝑥 = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdcnul 15075 | . . 3 ⊢ BOUNDED ∅ | |
2 | 1 | bdss 15074 | . 2 ⊢ BOUNDED 𝑥 ⊆ ∅ |
3 | 0ss 3476 | . . 3 ⊢ ∅ ⊆ 𝑥 | |
4 | eqss 3185 | . . 3 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥)) | |
5 | 3, 4 | mpbiran2 943 | . 2 ⊢ (𝑥 = ∅ ↔ 𝑥 ⊆ ∅) |
6 | 2, 5 | bd0r 15035 | 1 ⊢ BOUNDED 𝑥 = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ⊆ wss 3144 ∅c0 3437 BOUNDED wbd 15022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-bd0 15023 ax-bdim 15024 ax-bdn 15027 ax-bdal 15028 ax-bdeq 15030 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 df-bdc 15051 |
This theorem is referenced by: bj-bd0el 15078 bj-nn0suc0 15160 |
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