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Mirrors > Home > MPE Home > Th. List > Mathboxes > harn0 | Structured version Visualization version GIF version |
Description: The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
Ref | Expression |
---|---|
harn0 | ⊢ (𝑆 ∈ 𝑉 → (har‘𝑆) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6237 | . . . 4 ⊢ ∅ ∈ On | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∅ ∈ On) |
3 | 0domg 8632 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∅ ≼ 𝑆) | |
4 | elharval 9015 | . . 3 ⊢ (∅ ∈ (har‘𝑆) ↔ (∅ ∈ On ∧ ∅ ≼ 𝑆)) | |
5 | 2, 3, 4 | sylanbrc 583 | . 2 ⊢ (𝑆 ∈ 𝑉 → ∅ ∈ (har‘𝑆)) |
6 | 5 | ne0d 4298 | 1 ⊢ (𝑆 ∈ 𝑉 → (har‘𝑆) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 Oncon0 6184 ‘cfv 6348 ≼ cdom 8495 harchar 9008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-wrecs 7936 df-recs 7997 df-en 8498 df-dom 8499 df-oi 8962 df-har 9010 |
This theorem is referenced by: isnumbasabl 39584 dfacbasgrp 39586 |
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