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Mirrors > Home > MPE Home > Th. List > snnen2oOLD | Structured version Visualization version GIF version |
Description: Obsolete version of snnen2o 9258 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 8657 | . . . 4 ⊢ 1o ∈ ω | |
2 | php5 9235 | . . . 4 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ 1o ≈ suc 1o |
4 | ensn1g 9040 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
5 | df-2o 8484 | . . . . . 6 ⊢ 2o = suc 1o | |
6 | 5 | eqcomi 2734 | . . . . 5 ⊢ suc 1o = 2o |
7 | 6 | breq2i 5151 | . . . 4 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o) |
8 | ensymb 9019 | . . . . . 6 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴}) | |
9 | entr 9023 | . . . . . . 7 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o) | |
10 | 9 | ex 411 | . . . . . 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 329 | . . 3 ⊢ (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
14 | 3, 4, 13 | mpsyl 68 | . 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
15 | 2on0 8499 | . . . 4 ⊢ 2o ≠ ∅ | |
16 | ensymb 9019 | . . . . 5 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅) | |
17 | en0 9034 | . . . . 5 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
18 | 16, 17 | bitri 274 | . . . 4 ⊢ (∅ ≈ 2o ↔ 2o = ∅) |
19 | 15, 18 | nemtbir 3028 | . . 3 ⊢ ¬ ∅ ≈ 2o |
20 | snprc 4717 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
21 | 20 | biimpi 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
22 | 21 | breq1d 5153 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o)) |
23 | 19, 22 | mtbiri 326 | . 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 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 {csn 4624 class class class wbr 5143 suc csuc 6366 ωcom 7867 1oc1o 8476 2oc2o 8477 ≈ cen 8957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7868 df-1o 8483 df-2o 8484 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 |
This theorem is referenced by: (None) |
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