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Mirrors > Home > MPE Home > Th. List > snnen2o | Structured version Visualization version GIF version |
Description: A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.) |
Ref | Expression |
---|---|
snnen2o | ⊢ ¬ {𝐴} ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8005 | . . . 4 ⊢ 1o ∈ ω | |
2 | php5 8438 | . . . 4 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ 1o ≈ suc 1o |
4 | ensn1g 8308 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
5 | df-2o 7846 | . . . . . 6 ⊢ 2o = suc 1o | |
6 | 5 | eqcomi 2787 | . . . . 5 ⊢ suc 1o = 2o |
7 | 6 | breq2i 4896 | . . . 4 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o) |
8 | ensymb 8291 | . . . . . 6 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴}) | |
9 | entr 8295 | . . . . . . 7 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o) | |
10 | 9 | ex 403 | . . . . . 6 ⊢ (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
11 | 8, 10 | sylbi 209 | . . . . 5 ⊢ ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
12 | 11 | con3rr3 153 | . . . 4 ⊢ (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
13 | 7, 12 | sylnbi 322 | . . 3 ⊢ (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
14 | 3, 4, 13 | mpsyl 68 | . 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
15 | 2on0 7855 | . . . 4 ⊢ 2o ≠ ∅ | |
16 | ensymb 8291 | . . . . 5 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅) | |
17 | en0 8306 | . . . . 5 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
18 | 16, 17 | bitri 267 | . . . 4 ⊢ (∅ ≈ 2o ↔ 2o = ∅) |
19 | 15, 18 | nemtbir 3065 | . . 3 ⊢ ¬ ∅ ≈ 2o |
20 | snprc 4484 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
21 | 20 | biimpi 208 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
22 | 21 | breq1d 4898 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o)) |
23 | 19, 22 | mtbiri 319 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
24 | 14, 23 | pm2.61i 177 | 1 ⊢ ¬ {𝐴} ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 {csn 4398 class class class wbr 4888 suc csuc 5980 ωcom 7345 1oc1o 7838 2oc2o 7839 ≈ cen 8240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-1o 7845 df-2o 7846 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 |
This theorem is referenced by: pmtrsn 18327 |
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