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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eltag | Structured version Visualization version GIF version |
Description: Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-eltag | ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-tag 32156 | . . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅})) |
3 | elun 3715 | . 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅})) | |
4 | bj-elsngl 32149 | . . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥}) | |
5 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
6 | 5 | elsn2 4158 | . . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
7 | 4, 6 | orbi12i 542 | . 2 ⊢ ((𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}) ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
8 | 2, 3, 7 | 3bitri 285 | 1 ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∪ cun 3538 ∅c0 3874 {csn 4125 sngl bj-csngl 32146 tag bj-ctag 32155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 df-bj-sngl 32147 df-bj-tag 32156 |
This theorem is referenced by: (None) |
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