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Mirrors > Home > MPE Home > Th. List > Mathboxes > nolt02olem | Structured version Visualization version GIF version |
Description: Lemma for nolt02o 33203. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴 ⊆ 𝑋. (Contributed by Scott Fenton, 6-Dec-2021.) |
Ref | Expression |
---|---|
nolt02olem | ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nosgnn0 33169 | . . . 4 ⊢ ¬ ∅ ∈ {1o, 2o} | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → ¬ ∅ ∈ {1o, 2o}) |
3 | simpl3 1189 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) = ∅) | |
4 | simpl1 1187 | . . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → 𝐴 ∈ No ) | |
5 | norn 33162 | . . . . . 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 33160 | . . . . . . 7 ⊢ (𝐴 ∈ No → Fun 𝐴) | |
8 | 7 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → Fun 𝐴) |
9 | fvelrn 6847 | . . . . . 6 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴) | |
10 | 8, 9 | sylan 582 | . . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴) |
11 | 6, 10 | sseldd 3971 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ {1o, 2o}) |
12 | 3, 11 | eqeltrrd 2917 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) ∧ 𝑋 ∈ dom 𝐴) → ∅ ∈ {1o, 2o}) |
13 | 2, 12 | mtand 814 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → ¬ 𝑋 ∈ dom 𝐴) |
14 | nodmon 33161 | . . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) | |
15 | 14 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ∈ On) |
16 | simp2 1133 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → 𝑋 ∈ On) | |
17 | ontri1 6228 | . . 3 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ On) → (dom 𝐴 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴)) | |
18 | 15, 16, 17 | syl2anc 586 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → (dom 𝐴 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴)) |
19 | 13, 18 | mpbird 259 | 1 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ∅c0 4294 {cpr 4572 dom cdm 5558 ran crn 5559 Oncon0 6194 Fun wfun 6352 ‘cfv 6358 1oc1o 8098 2oc2o 8099 No csur 33151 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1o 8105 df-2o 8106 df-no 33154 |
This theorem is referenced by: nolt02o 33203 |
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