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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sngltag | Structured version Visualization version GIF version |
Description: The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-sngltag | ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-sngltagi 34191 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵) | |
2 | df-bj-tag 34184 | . . . 4 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
3 | 2 | eleq2i 2901 | . . 3 ⊢ ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅})) |
4 | elun 4122 | . . . 4 ⊢ ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅})) | |
5 | idd 24 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵)) | |
6 | elsni 4574 | . . . . . 6 ⊢ ({𝐴} ∈ {∅} → {𝐴} = ∅) | |
7 | snprc 4645 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | elex 3510 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
9 | 8 | pm2.24d 154 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵)) |
10 | 7, 9 | syl5bir 244 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵)) |
11 | 6, 10 | syl5 34 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵)) |
12 | 5, 11 | jaod 853 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵)) |
13 | 4, 12 | syl5bi 243 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵)) |
14 | 3, 13 | syl5bi 243 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵)) |
15 | 1, 14 | impbid2 227 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∅c0 4288 {csn 4557 sngl bj-csngl 34174 tag bj-ctag 34183 |
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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-bj-tag 34184 |
This theorem is referenced by: bj-tagcg 34194 bj-taginv 34195 |
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