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Mirrors > Home > ILE Home > Th. List > snnen2og | GIF version |
Description: A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a proper class, see snnen2oprc 6395. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
snnen2og | ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ≈ 2𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6159 | . . 3 ⊢ 1𝑜 ∈ ω | |
2 | php5 6393 | . . 3 ⊢ (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ ¬ 1𝑜 ≈ suc 1𝑜 |
4 | ensn1g 6344 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) | |
5 | df-2o 6066 | . . . . 5 ⊢ 2𝑜 = suc 1𝑜 | |
6 | 5 | eqcomi 2086 | . . . 4 ⊢ suc 1𝑜 = 2𝑜 |
7 | 6 | breq2i 3801 | . . 3 ⊢ (1𝑜 ≈ suc 1𝑜 ↔ 1𝑜 ≈ 2𝑜) |
8 | ensymb 6327 | . . . . 5 ⊢ ({𝐴} ≈ 1𝑜 ↔ 1𝑜 ≈ {𝐴}) | |
9 | entr 6331 | . . . . . 6 ⊢ ((1𝑜 ≈ {𝐴} ∧ {𝐴} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜) | |
10 | 9 | ex 113 | . . . . 5 ⊢ (1𝑜 ≈ {𝐴} → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜)) |
11 | 8, 10 | sylbi 119 | . . . 4 ⊢ ({𝐴} ≈ 1𝑜 → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜)) |
12 | 11 | con3rr3 596 | . . 3 ⊢ (¬ 1𝑜 ≈ 2𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜)) |
13 | 7, 12 | sylnbi 636 | . 2 ⊢ (¬ 1𝑜 ≈ suc 1𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜)) |
14 | 3, 4, 13 | mpsyl 64 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ≈ 2𝑜) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1434 {csn 3406 class class class wbr 3793 suc csuc 4128 ωcom 4339 1𝑜c1o 6058 2𝑜c2o 6059 ≈ cen 6285 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-br 3794 df-opab 3848 df-tr 3884 df-id 4056 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-1o 6065 df-2o 6066 df-er 6172 df-en 6288 |
This theorem is referenced by: (None) |
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