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Theorem nnsdomgOLD 9244
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.)
Assertion
Ref Expression
nnsdomgOLD ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

Proof of Theorem nnsdomgOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef 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)
42, 3mpan 688 . . 3 (𝐴 ∈ ω → 𝐴 ⊆ ω)
51, 4impel 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))
98rspcev 3580 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
10 isfi 8913 . . . . . . 7 (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
119, 10sylibr 233 . . . . . 6 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
1211ex 413 . . . . 5 (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
137, 12syl5 34 . . . 4 (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
146, 13mtoi 198 . . 3 (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
1514adantl 482 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
16 brsdom 8912 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
175, 15, 16sylanbrc 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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