Theorem nolt02olem 27658

Description: Lemma for nolt02o 27659. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴 ⊆ 𝑋. (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nolt02olem	⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋)

Proof of Theorem nolt02olem

Step	Hyp	Ref	Expression
1		nosgnn0 27622	. . . 4 ⊢ ¬ ∅ ∈ {1o, 2o}
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → ¬ ∅ ∈ {1o, 2o})
3		simpl3 1195	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) = ∅)
4		simpl1 1193	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → 𝐴 ∈ No )
5		norn 27615	. . . . . 6 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
6	4, 5	syl 17	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → ran 𝐴 ⊆ {1o, 2o})
7		nofun 27613	. . . . . . 7 ⊢ (𝐴 ∈ No → Fun 𝐴)
8	7	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → Fun 𝐴)
9		fvelrn 7028	. . . . . 6 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴)
10	8, 9	sylan 581	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴)
11	6, 10	sseldd 3922	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ {1o, 2o})
12	3, 11	eqeltrrd 2837	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → ∅ ∈ {1o, 2o})
13	2, 12	mtand 816	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → ¬ 𝑋 ∈ dom 𝐴)
14		nodmon 27614	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
15	14	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ∈ On)
16		simp2 1138	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → 𝑋 ∈ On)
17		ontri1 6357	. . 3 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ On) → (dom 𝐴 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴))
18	15, 16, 17	syl2anc 585	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → (dom 𝐴 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴))
19	13, 18	mpbird 257	1 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ∅c0 4273  {cpr 4569  dom cdm 5631  ran crn 5632  Oncon0 6323  Fun wfun 6492  ‘cfv 6498  1_oc1o 8398  2_oc2o 8399   No csur 27603

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-1o 8405  df-2o 8406  df-no 27606

This theorem is referenced by:  nolt02o  27659  nogt01o  27660