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| Mirrors > Home > MPE Home > Th. List > nnsdomgOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of nnsdomg 9335 as of 7-Jan-2025. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nnsdomgOLD | ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssdomg 9040 | . . 3 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
| 2 | ordom 7897 | . . . 4 ⊢ Ord ω | |
| 3 | ordelss 6400 | . . . 4 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
| 4 | 2, 3 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
| 5 | 1, 4 | impel 505 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω) |
| 6 | ominf 9294 | . . . 4 ⊢ ¬ ω ∈ Fin | |
| 7 | ensym 9043 | . . . . 5 ⊢ (𝐴 ≈ ω → ω ≈ 𝐴) | |
| 8 | breq2 5147 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴)) | |
| 9 | 8 | rspcev 3622 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥) |
| 10 | isfi 9016 | . . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
| 11 | 9, 10 | sylibr 234 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin) |
| 12 | 11 | ex 412 | . . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin)) |
| 13 | 7, 12 | syl5 34 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin)) |
| 14 | 6, 13 | mtoi 199 | . . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω) |
| 15 | 14 | adantl 481 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω) |
| 16 | brsdom 9015 | . 2 ⊢ (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω)) | |
| 17 | 5, 15, 16 | sylanbrc 583 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 Ord word 6383 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 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-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-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 |
| This theorem is referenced by: (None) |
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