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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 6873. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
snnen2og | ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6534 | . . 3 ⊢ 1o ∈ ω | |
2 | php5 6871 | . . 3 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ¬ 1o ≈ suc 1o |
4 | ensn1g 6810 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
5 | df-2o 6431 | . . . . 5 ⊢ 2o = suc 1o | |
6 | 5 | eqcomi 2191 | . . . 4 ⊢ suc 1o = 2o |
7 | 6 | breq2i 4023 | . . 3 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o) |
8 | ensymb 6793 | . . . . 5 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴}) | |
9 | entr 6797 | . . . . . 6 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o) | |
10 | 9 | ex 115 | . . . . 5 ⊢ (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
11 | 8, 10 | sylbi 121 | . . . 4 ⊢ ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
12 | 11 | con3rr3 634 | . . 3 ⊢ (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
13 | 7, 12 | sylnbi 679 | . 2 ⊢ (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
14 | 3, 4, 13 | mpsyl 65 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ≈ 2o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2158 {csn 3604 class class class wbr 4015 suc csuc 4377 ωcom 4601 1oc1o 6423 2oc2o 6424 ≈ cen 6751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-1o 6430 df-2o 6431 df-er 6548 df-en 6754 |
This theorem is referenced by: (None) |
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