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Mirrors > Home > MPE Home > Th. List > nnsdomgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of nnsdomg 9332 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 9038 | . . 3 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
2 | ordom 7896 | . . . 4 ⊢ Ord ω | |
3 | ordelss 6401 | . . . 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 9291 | . . . 4 ⊢ ¬ ω ∈ Fin | |
7 | ensym 9041 | . . . . 5 ⊢ (𝐴 ≈ ω → ω ≈ 𝐴) | |
8 | breq2 5151 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴)) | |
9 | 8 | rspcev 3621 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥) |
10 | isfi 9014 | . . . . . . 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 9013 | . 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 2105 ∃wrex 3067 Vcvv 3477 ⊆ wss 3962 class class class wbr 5147 Ord word 6384 ωcom 7886 ≈ cen 8980 ≼ cdom 8981 ≺ csdm 8982 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-om 7887 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 |
This theorem is referenced by: (None) |
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