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| Mirrors > Home > MPE Home > Th. List > snfiOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of snfi 9083 as of 13-Jan-2025. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| snfiOLD | ⊢ {𝐴} ∈ Fin |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1onn 8678 | . . . 4 ⊢ 1o ∈ ω | |
| 2 | ensn1g 9062 | . . . 4 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
| 3 | breq2 5147 | . . . . 5 ⊢ (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o)) | |
| 4 | 3 | rspcev 3622 | . . . 4 ⊢ ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
| 5 | 1, 2, 4 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
| 6 | snprc 4717 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 7 | en0 9058 | . . . . 5 ⊢ ({𝐴} ≈ ∅ ↔ {𝐴} = ∅) | |
| 8 | peano1 7910 | . . . . . 6 ⊢ ∅ ∈ ω | |
| 9 | breq2 5147 | . . . . . . 7 ⊢ (𝑥 = ∅ → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ ∅)) | |
| 10 | 9 | rspcev 3622 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ {𝐴} ≈ ∅) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
| 11 | 8, 10 | mpan 690 | . . . . 5 ⊢ ({𝐴} ≈ ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
| 12 | 7, 11 | sylbir 235 | . . . 4 ⊢ ({𝐴} = ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
| 13 | 6, 12 | sylbi 217 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
| 14 | 5, 13 | pm2.61i 182 | . 2 ⊢ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥 |
| 15 | isfi 9016 | . 2 ⊢ ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) | |
| 16 | 14, 15 | mpbir 231 | 1 ⊢ {𝐴} ∈ Fin |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∅c0 4333 {csn 4626 class class class wbr 5143 ωcom 7887 1oc1o 8499 ≈ cen 8982 Fincfn 8985 |
| 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-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 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-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-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-om 7888 df-1o 8506 df-en 8986 df-fin 8989 |
| This theorem is referenced by: (None) |
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