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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 8550 | . . . . 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 4049 | . 2 ⊢ (𝑋 ≼ 𝑌 → {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋} ⊆ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑌}) |
5 | reldom 8503 | . . . 4 ⊢ Rel ≼ | |
6 | 5 | brrelex1i 5601 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ∈ V) |
7 | harval 9014 | . . 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 5602 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
10 | harval 9014 | . . 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 4011 | 1 ⊢ (𝑋 ≼ 𝑌 → (har‘𝑋) ⊆ (har‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ⊆ wss 3933 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: hsmexlem3 9838 |
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