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Mirrors > Home > ILE Home > Th. List > snnen2oprc | GIF version |
Description: A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a set, see snnen2og 6761. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
snnen2oprc | ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 6331 | . . 3 ⊢ 2o ≠ ∅ | |
2 | ensymb 6682 | . . . 4 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅) | |
3 | en0 6697 | . . . 4 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
4 | 2, 3 | bitri 183 | . . 3 ⊢ (∅ ≈ 2o ↔ 2o = ∅) |
5 | 1, 4 | nemtbir 2398 | . 2 ⊢ ¬ ∅ ≈ 2o |
6 | snprc 3596 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
7 | 6 | biimpi 119 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
8 | 7 | breq1d 3947 | . 2 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o)) |
9 | 5, 8 | mtbiri 665 | 1 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∅c0 3368 {csn 3532 class class class wbr 3937 2oc2o 6315 ≈ cen 6640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-1o 6321 df-2o 6322 df-er 6437 df-en 6643 |
This theorem is referenced by: (None) |
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