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Mirrors > Home > MPE Home > Th. List > harword | Structured version Visualization version GIF version |
Description: Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
harword | ⊢ (𝑋 ≼ 𝑌 → (har‘𝑋) ⊆ (har‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domtr 8905 | . . . . 5 ⊢ ((𝑦 ≼ 𝑋 ∧ 𝑋 ≼ 𝑌) → 𝑦 ≼ 𝑌) | |
2 | 1 | expcom 414 | . . . 4 ⊢ (𝑋 ≼ 𝑌 → (𝑦 ≼ 𝑋 → 𝑦 ≼ 𝑌)) |
3 | 2 | adantr 481 | . . 3 ⊢ ((𝑋 ≼ 𝑌 ∧ 𝑦 ∈ On) → (𝑦 ≼ 𝑋 → 𝑦 ≼ 𝑌)) |
4 | 3 | ss2rabdv 4031 | . 2 ⊢ (𝑋 ≼ 𝑌 → {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋} ⊆ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑌}) |
5 | reldom 8847 | . . . 4 ⊢ Rel ≼ | |
6 | 5 | brrelex1i 5686 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ∈ V) |
7 | harval 9454 | . . 3 ⊢ (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝑋 ≼ 𝑌 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}) |
9 | 5 | brrelex2i 5687 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
10 | harval 9454 | . . 3 ⊢ (𝑌 ∈ V → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑌}) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝑋 ≼ 𝑌 → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑌}) |
12 | 4, 8, 11 | 3sstr4d 3989 | 1 ⊢ (𝑋 ≼ 𝑌 → (har‘𝑋) ⊆ (har‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3405 Vcvv 3443 ⊆ wss 3908 class class class wbr 5103 Oncon0 6315 ‘cfv 6493 ≼ cdom 8839 harchar 9450 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-en 8842 df-dom 8843 df-oi 9404 df-har 9451 |
This theorem is referenced by: hsmexlem3 10322 |
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