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Mirrors > Home > MPE Home > Th. List > nnsdomgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of nnsdomg 9243 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 8937 | . . 3 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
2 | ordom 7809 | . . . 4 ⊢ Ord ω | |
3 | ordelss 6332 | . . . 4 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
5 | 1, 4 | impel 506 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω) |
6 | ominf 9199 | . . . 4 ⊢ ¬ ω ∈ Fin | |
7 | ensym 8940 | . . . . 5 ⊢ (𝐴 ≈ ω → ω ≈ 𝐴) | |
8 | breq2 5108 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴)) | |
9 | 8 | rspcev 3580 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥) |
10 | isfi 8913 | . . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
11 | 9, 10 | sylibr 233 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin) |
12 | 11 | ex 413 | . . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin)) |
13 | 7, 12 | syl5 34 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin)) |
14 | 6, 13 | mtoi 198 | . . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω) |
15 | 14 | adantl 482 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω) |
16 | brsdom 8912 | . 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 396 ∈ wcel 2106 ∃wrex 3072 Vcvv 3444 ⊆ wss 3909 class class class wbr 5104 Ord word 6315 ωcom 7799 ≈ cen 8877 ≼ cdom 8878 ≺ csdm 8879 Fincfn 8880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-om 7800 df-1o 8409 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 |
This theorem is referenced by: (None) |
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