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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 6596 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
2 | id 19 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
3 | unieq 3684 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | vex 2636 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | unisn 3691 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 |
6 | 3, 5 | syl6eq 2143 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) |
7 | 6 | sneqd 3479 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) |
8 | 2, 7 | eqtr4d 2130 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
9 | 8 | exlimiv 1541 | . . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) |
10 | 1, 9 | sylbi 120 | . 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴}) |
11 | uniexg 4290 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
12 | ensn1g 6594 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o) | |
13 | 11, 12 | syl 14 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∪ 𝐴} ≈ 1o) |
14 | breq1 3870 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → (𝐴 ≈ 1o ↔ {∪ 𝐴} ≈ 1o)) | |
15 | 13, 14 | syl5ibrcom 156 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o)) |
16 | 10, 15 | impbid2 142 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1296 ∃wex 1433 ∈ wcel 1445 Vcvv 2633 {csn 3466 ∪ cuni 3675 class class class wbr 3867 1oc1o 6212 ≈ cen 6535 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-suc 4222 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-1o 6219 df-en 6538 |
This theorem is referenced by: en1uniel 6601 |
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