| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > snnen2oOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of snnen2o 9273 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 8678 | . . . 4 ⊢ 1o ∈ ω | |
| 2 | php5 9251 | . . . 4 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ 1o ≈ suc 1o |
| 4 | ensn1g 9062 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
| 5 | df-2o 8507 | . . . . . 6 ⊢ 2o = suc 1o | |
| 6 | 5 | eqcomi 2746 | . . . . 5 ⊢ suc 1o = 2o |
| 7 | 6 | breq2i 5151 | . . . 4 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o) |
| 8 | ensymb 9042 | . . . . . 6 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴}) | |
| 9 | entr 9046 | . . . . . . 7 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o) | |
| 10 | 9 | ex 412 | . . . . . 6 ⊢ (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
| 11 | 8, 10 | sylbi 217 | . . . . 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 8522 | . . . 4 ⊢ 2o ≠ ∅ | |
| 16 | ensymb 9042 | . . . . 5 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅) | |
| 17 | en0 9058 | . . . . 5 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
| 18 | 16, 17 | bitri 275 | . . . 4 ⊢ (∅ ≈ 2o ↔ 2o = ∅) |
| 19 | 15, 18 | nemtbir 3038 | . . 3 ⊢ ¬ ∅ ≈ 2o |
| 20 | snprc 4717 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 21 | 20 | biimpi 216 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 22 | 21 | breq1d 5153 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 class class class wbr 5143 suc csuc 6386 ωcom 7887 1oc1o 8499 2oc2o 8500 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |