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Mirrors > Home > MPE Home > Th. List > en1bOLD | Structured version Visualization version GIF version |
Description: Obsolete version of en1b 8678 as of 24-Sep-2024. (Contributed by Mario Carneiro, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
en1bOLD | ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 8676 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 4816 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | vex 3402 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | unisn 4827 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 |
6 | 3, 5 | eqtrdi 2787 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
7 | 6 | sneqd 4539 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
8 | 2, 7 | eqtr4d 2774 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
9 | 8 | exlimiv 1938 | . . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
10 | 1, 9 | sylbi 220 | . 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴}) |
11 | id 22 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴}) | |
12 | snex 5309 | . . . . . 6 ⊢ {∪ 𝐴} ∈ V | |
13 | 11, 12 | eqeltrdi 2839 | . . . . 5 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ∈ V) |
14 | 13 | uniexd 7508 | . . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) |
15 | ensn1g 8674 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o) |
17 | 11, 16 | eqbrtrd 5061 | . 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o) |
18 | 10, 17 | impbii 212 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∃wex 1787 ∈ wcel 2112 Vcvv 3398 {csn 4527 ∪ cuni 4805 class class class wbr 5039 1oc1o 8173 ≈ cen 8601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1o 8180 df-en 8605 |
This theorem is referenced by: (None) |
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