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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 32963 | . . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
2 | 1 | eleq2i 2693 | . 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅})) |
3 | elun 3753 | . 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅})) | |
4 | bj-elsngl 32956 | . . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥}) | |
5 | 0ex 4790 | . . . 4 ⊢ ∅ ∈ V | |
6 | 5 | elsn2 4211 | . . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅) |
7 | 4, 6 | orbi12i 543 | . 2 ⊢ ((𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}) ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
8 | 2, 3, 7 | 3bitri 286 | 1 ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∪ cun 3572 ∅c0 3915 {csn 4177 sngl bj-csngl 32953 tag bj-ctag 32962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 df-bj-sngl 32954 df-bj-tag 32963 |
This theorem is referenced by: (None) |
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