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Mirrors > Home > MPE Home > Th. List > en1b | Structured version Visualization version GIF version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) |
Ref | Expression |
---|---|
en1b | ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 8559 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 4811 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | vex 3444 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | unisn 4820 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 |
6 | 3, 5 | eqtrdi 2849 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
7 | 6 | sneqd 4537 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
8 | 2, 7 | eqtr4d 2836 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
9 | 8 | exlimiv 1931 | . . 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 5297 | . . . . . 6 ⊢ {∪ 𝐴} ∈ V | |
13 | 11, 12 | eqeltrdi 2898 | . . . . 5 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ∈ V) |
14 | 13 | uniexd 7448 | . . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) |
15 | ensn1g 8557 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o) |
17 | 11, 16 | eqbrtrd 5052 | . 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 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 {csn 4525 ∪ cuni 4800 class class class wbr 5030 1oc1o 8078 ≈ cen 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-1o 8085 df-en 8493 |
This theorem is referenced by: en1uniel 8564 sylow2alem2 18735 sylow2a 18736 frgpcyg 20265 ptcmplem3 22659 cnextfvval 22670 cnextcn 22672 minveclem4a 24034 isppw 25699 xrge0tsmsbi 30743 |
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