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Mirrors > Home > MPE Home > Th. List > 0sdom1domALT | Structured version Visualization version GIF version |
Description: Alternate proof of 0sdom1dom 9272, shorter but requiring ax-un 7754. (Contributed by NM, 28-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0sdom1domALT | ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7911 | . . 3 ⊢ ∅ ∈ ω | |
2 | sucdom 9269 | . . 3 ⊢ (∅ ∈ ω → (∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴) |
4 | df-1o 8505 | . . 3 ⊢ 1o = suc ∅ | |
5 | 4 | breq1i 5155 | . 2 ⊢ (1o ≼ 𝐴 ↔ suc ∅ ≼ 𝐴) |
6 | 3, 5 | bitr4i 278 | 1 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 suc csuc 6388 ωcom 7887 1oc1o 8498 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 |
This theorem is referenced by: (None) |
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