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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 6449 | . . 3 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 19 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 3639 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | vex 2617 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | unisn 3646 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 |
6 | 3, 5 | syl6eq 2133 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
7 | 6 | sneqd 3438 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
8 | 2, 7 | eqtr4d 2120 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
9 | 8 | exlimiv 1532 | . . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
10 | 1, 9 | sylbi 119 | . 2 ⊢ (𝐴 ≈ 1𝑜 → 𝐴 = {∪ 𝐴}) |
11 | uniexg 4232 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
12 | ensn1g 6447 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1𝑜) | |
13 | 11, 12 | syl 14 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∪ 𝐴} ≈ 1𝑜) |
14 | breq1 3817 | . . 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 1287 ∃wex 1424 ∈ wcel 1436 Vcvv 2614 {csn 3425 ∪ cuni 3630 class class class wbr 3814 1𝑜c1o 6109 ≈ cen 6388 |
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 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-reu 2362 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-id 4087 df-suc 4165 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-1o 6116 df-en 6391 |
This theorem is referenced by: en1uniel 6454 |
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