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Mirrors > Home > MPE Home > Th. List > snnen2oOLD | Structured version Visualization version GIF version |
Description: Obsolete version of snnen2o 9188 as of 18-Nov-2024. (Contributed by AV, 6-Aug-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
snnen2oOLD | ⊢ ¬ {𝐴} ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8591 | . . . 4 ⊢ 1o ∈ ω | |
2 | php5 9165 | . . . 4 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ 1o ≈ suc 1o |
4 | ensn1g 8970 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
5 | df-2o 8418 | . . . . . 6 ⊢ 2o = suc 1o | |
6 | 5 | eqcomi 2746 | . . . . 5 ⊢ suc 1o = 2o |
7 | 6 | breq2i 5118 | . . . 4 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o) |
8 | ensymb 8949 | . . . . . 6 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴}) | |
9 | entr 8953 | . . . . . . 7 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o) | |
10 | 9 | ex 414 | . . . . . 6 ⊢ (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
11 | 8, 10 | sylbi 216 | . . . . 5 ⊢ ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
12 | 11 | con3rr3 155 | . . . 4 ⊢ (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
13 | 7, 12 | sylnbi 330 | . . 3 ⊢ (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
14 | 3, 4, 13 | mpsyl 68 | . 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
15 | 2on0 8433 | . . . 4 ⊢ 2o ≠ ∅ | |
16 | ensymb 8949 | . . . . 5 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅) | |
17 | en0 8964 | . . . . 5 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
18 | 16, 17 | bitri 275 | . . . 4 ⊢ (∅ ≈ 2o ↔ 2o = ∅) |
19 | 15, 18 | nemtbir 3041 | . . 3 ⊢ ¬ ∅ ≈ 2o |
20 | snprc 4683 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
21 | 20 | biimpi 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
22 | 21 | breq1d 5120 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o)) |
23 | 19, 22 | mtbiri 327 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
24 | 14, 23 | pm2.61i 182 | 1 ⊢ ¬ {𝐴} ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∅c0 4287 {csn 4591 class class class wbr 5110 suc csuc 6324 ωcom 7807 1oc1o 8410 2oc2o 8411 ≈ cen 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-2o 8418 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 |
This theorem is referenced by: (None) |
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