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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 7721. (Revised by BTernaryTau, 24-Sep-2024.) |
Ref | Expression |
---|---|
en1b | ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 9017 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 4918 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | unisnv 4930 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 | |
5 | 3, 4 | eqtrdi 2788 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
6 | 5 | sneqd 4639 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
7 | 2, 6 | eqtr4d 2775 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
8 | 7 | exlimiv 1933 | . . 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 5358 | . . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) | |
12 | ensn1g 9015 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o) |
14 | 10, 13 | eqbrtrd 5169 | . 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 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 {csn 4627 ∪ cuni 4907 class class class wbr 5147 1oc1o 8455 ≈ cen 8932 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1o 8462 df-en 8936 |
This theorem is referenced by: en1uniel 9024 en1unielOLD 9025 sylow2alem2 19480 sylow2a 19481 frgpcyg 21120 ptcmplem3 23549 cnextfvval 23560 cnextcn 23562 minveclem4a 24938 isppw 26607 xrge0tsmsbi 32197 |
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