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Mirrors > Home > MPE Home > Th. List > wdomac | Structured version Visualization version GIF version |
Description: When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
wdomac | ⊢ (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 9503 | . . 3 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5690 | . 2 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | reldom 8890 | . . 3 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5690 | . 2 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
5 | numth3 10407 | . . 3 ⊢ (𝑌 ∈ V → 𝑌 ∈ dom card) | |
6 | wdomnumr 10001 | . . 3 ⊢ (𝑌 ∈ dom card → (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌)) |
8 | 2, 4, 7 | pm5.21nii 380 | 1 ⊢ (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2107 Vcvv 3446 class class class wbr 5106 dom cdm 5634 ≼ cdom 8882 ≼* cwdom 9501 cardccrd 9872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-ac2 10400 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-wdom 9502 df-card 9876 df-acn 9879 df-ac 10053 |
This theorem is referenced by: (None) |
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