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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.) Avoid ax-un 7677. (Revised by BTernaryTau, 24-Sep-2024.) |
Ref | Expression |
---|---|
en1b | ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 8972 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 4881 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | unisnv 4893 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 | |
5 | 3, 4 | eqtrdi 2793 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
6 | 5 | sneqd 4603 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
7 | 2, 6 | eqtr4d 2780 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
8 | 7 | exlimiv 1934 | . . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
9 | 1, 8 | sylbi 216 | . 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴}) |
10 | id 22 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴}) | |
11 | eqsnuniex 5321 | . . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) | |
12 | ensn1g 8970 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o) |
14 | 10, 13 | eqbrtrd 5132 | . 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o) |
15 | 9, 14 | impbii 208 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3448 {csn 4591 ∪ cuni 4870 class class class wbr 5110 1oc1o 8410 ≈ cen 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-1o 8417 df-en 8891 |
This theorem is referenced by: en1uniel 8979 en1unielOLD 8980 sylow2alem2 19407 sylow2a 19408 frgpcyg 20996 ptcmplem3 23421 cnextfvval 23432 cnextcn 23434 minveclem4a 24810 isppw 26479 xrge0tsmsbi 31942 |
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