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Mirrors > Home > MPE Home > Th. List > 1sdom2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of 1sdom2 9181, shorter but requiring ax-un 7669. (Contributed by NM, 4-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1sdom2ALT | ⊢ 1o ≺ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8583 | . . 3 ⊢ 1o ∈ ω | |
2 | php4 9154 | . . 3 ⊢ (1o ∈ ω → 1o ≺ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 1o ≺ suc 1o |
4 | df-2o 8410 | . 2 ⊢ 2o = suc 1o | |
5 | 3, 4 | breqtrri 5131 | 1 ⊢ 1o ≺ 2o |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 class class class wbr 5104 suc csuc 6318 ωcom 7799 1oc1o 8402 2oc2o 8403 ≺ csdm 8879 |
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 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7669 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-om 7800 df-1o 8409 df-2o 8410 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 |
This theorem is referenced by: (None) |
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