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Mirrors > Home > ILE Home > Th. List > en1bg | GIF version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.) |
Ref | Expression |
---|---|
en1bg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ 1𝑜 ↔ 𝐴 = {∪ 𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 6302 | . . 3 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 19 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 3610 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | vex 2604 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | unisn 3617 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 |
6 | 3, 5 | syl6eq 2129 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
7 | 6 | sneqd 3411 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
8 | 2, 7 | eqtr4d 2116 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
9 | 8 | exlimiv 1529 | . . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
10 | 1, 9 | sylbi 119 | . 2 ⊢ (𝐴 ≈ 1𝑜 → 𝐴 = {∪ 𝐴}) |
11 | uniexg 4193 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
12 | ensn1g 6300 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1𝑜) | |
13 | 11, 12 | syl 14 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∪ 𝐴} ≈ 1𝑜) |
14 | breq1 3788 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → (𝐴 ≈ 1𝑜 ↔ {∪ 𝐴} ≈ 1𝑜)) | |
15 | 13, 14 | syl5ibrcom 155 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1𝑜)) |
16 | 10, 15 | impbid2 141 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ 1𝑜 ↔ 𝐴 = {∪ 𝐴})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 {csn 3398 ∪ cuni 3601 class class class wbr 3785 1𝑜c1o 6017 ≈ cen 6242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-suc 4126 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1o 6024 df-en 6245 |
This theorem is referenced by: en1uniel 6307 |
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