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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 36518 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵) | |
2 | df-bj-tag 36511 | . . . 4 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
3 | 2 | eleq2i 2817 | . . 3 ⊢ ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅})) |
4 | elun 4141 | . . . 4 ⊢ ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅})) | |
5 | idd 24 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵)) | |
6 | elsni 4641 | . . . . . 6 ⊢ ({𝐴} ∈ {∅} → {𝐴} = ∅) | |
7 | snprc 4717 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | elex 3482 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
9 | 8 | pm2.24d 151 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵)) |
10 | 7, 9 | biimtrrid 242 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵)) |
11 | 6, 10 | syl5 34 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵)) |
12 | 5, 11 | jaod 857 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵)) |
13 | 4, 12 | biimtrid 241 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵)) |
14 | 3, 13 | biimtrid 241 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵)) |
15 | 1, 14 | impbid2 225 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∪ cun 3937 ∅c0 4318 {csn 4624 sngl bj-csngl 36501 tag bj-ctag 36510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-sn 4625 df-bj-tag 36511 |
This theorem is referenced by: bj-tagcg 36521 bj-taginv 36522 |
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